Digitized by the Internet Archive
in 2011 with funding from
Research Library, The Getty Research Institute
http://www.archive.org/details/firstsixbooksofeOObyrn
C'€>r>-'^
~?
'ih
BYRNE'S EUCLID
THE FIRST SIX BOOKS OF
liTHE ELEMENTS OF EUCLID
WITH COLOURED DIAGRAMS
AND SYMBOLS
.V' »
THE FIRST SIX BOOKS OF
THE ELEMENTS OF EUCLID
IN WHICH COLOURED DIAGRAMS AND SYMBOLS
ARE USED INSTEAD OF LETTERS FOR THE
GREATER EASE OF LEARNERS
BY OLIVER BYRNE
SURVEYOR OF HER MAJESTY'S SETTLEMENTS IN THE FALKLAND ISLANDS
AND AUTHOR OF NUMEROUS MATHEMATICAL WORKS
LONDON
WILLIAM PICKERING
1847
TO THE
RIGHT HONOURABLE THE EARL FITZWILLL\M,
ETC. ETC. ETC.
THIS WORK IS DEDICATED
BY HIS LORDSHIPS OBEDIENT
AND MUCH OBLIGED SERVANT,
OLIVER BYRNE.
INTRODUCTION.
HE arts and fciences have become fo extenfive,
that to faciUtate their acquirement is of as
much importance as to extend their boundaries.
Illuftration, if it does not fhorten the time of
ftudy, will at leaft make it more agreeable. This Work
has a greater aim than mere illuftration ; we do not intro-
duce colours for the purpofe of entertainment, or to amufe
by certain combinations of tint and form, but to airift the
mind in its refearches after truth, to increafe the facilities
of inflrudlion, and to diffufe permanent knowledge. If we
wanted authorities to prove the importance and ufefulnefs
of geometry, we might quote every philofopher fmce the
days of Plato. Among the Greeks, in ancient, as in the
fchool of Peftalozzi and others in recent times, geometry
was adopted as the befl: gymnaftic of the mind. In facfl,
Euclid's Elements have become, by common confent, the
bafis of mathematical fcience all over the civilized globe.
But this will not appear extraordinary, if we confider that
this fublime fcience is not only better calculated than any
other to call forth the fpirit of inquiry, to elevate the mind,
and to ftrengthen the reafoning faculties, but alfo it forms
the beft introdudlion to moft of the ufeful and important
vocations of human life. Arithmetic, land-furveying, men-
furation, engineering, navigation, mechanics, hydroftatics,
pneumatics, optics, phyfical aftronomy, &c. are all depen-
dent on the propolitions of geometry.
viii INTRODUCTION.
Much however depends on the firft communication of
any fcience to a learner, though the beft and moft eafy
methods are feldom adopted. Propofitions are placed be-
fore a ftudent, who though having a fufficient underftand-
ing, is told juft as much about them on entering at the
very threfliold of the fcience, as gives him a prepolleffion
moft unfavourable to his future ftudy of this delightful
fubjedl ; or " the formalities and paraphernalia of rigour are
fo oftentatioufly put forward, as almoft to hide the reality.
Endlefs and perplexing repetitions, which do not confer
greater exactitude on the reafoning, render the demonftra-
tions involved and obfcure, and conceal from the view of
the ftudent the confecution of evidence." Thus an aver-
fion is created in the mind of the pupil, and a fubjeft fo
calculated to improve the reafoning powers, and give the
habit of clofe thinking, is degraded by a dry and rigid
courfe of inftrudlion into an uninterefting exercife of the
memory. To raife the curiofity, and to awaken the liftlefs
and dormant powers of younger minds fliould be the aim
of every teacher ; but where examples of excellence are
wanting, the attempts to attain it are but few, while emi-
nence excites attention and produces imitation. The objedl
of this Work is to introduce a method of teaching geome-
try, which has been much approved of by many fcientific
men in this country, as well as in France and America.
The plan here adopted forcibly appeals to the eye, the moft
fenlitive and the moft comprehenfive of our external organs,
and its pre-eminence to imprint it fubjedl on the mind is
fupported by the incontrovertible maxim exprefled in the
well known words of Horace : —
Segnius irritant animos demijfa per auran
^uam qua fimt oculis fuhjeSla fidelibus.
A feebler imprefs through the ear is made,
Than what is by the faithful eye conveyed.
INTRODUCTION. ix
All language confifts of reprefentative figns, and thole
figns are the befl which efFedl their purpofes with the
greateft precifion and difpatch. Such for all common pur-
pofes are the audible figns called words, which are ftill
confidered as audible, whether addreffed immediately to the
ear, or through the medium of letters to the eye. Geo-
metrical diagrams are not figns, but the materials of geo-
metrical fcience, the objedt of which is to Ihow the relative
quantities of their parts by a procefs of reafoning called
Demonftration. This reafoning has been generally carried
on by words, letters, and black or uncoloured diagrams ;
but as the ufe of coloured fymbols, figns, and diagrams in
the linear arts and fciences, renders the procefs of reafon-
ing more precife, and the attainment more expeditious, they
have been in this inflance accordingly adopted.
Such is the expedition of this enticing mode of commu-
nicating knowledge, that the Elements of Euclid can be
acquired in lefs than one third the time ufually employed,
and the retention by the memory is much more permanent;
thefe facts have been afcertained by numerous experiments
made by the inventor, and feveral others who have adopted
his plans. The particulars of which are few and obvious ;
the letters annexed to points, lines, or other parts of a dia-
gram are in fadt but arbitrary names, and reprefent them in
the demonftration ; inftead of thefe, the parts being differ-
ently coloured, are made g
to name themfelves, for
their forms incorrefpond-
ing colours represent them
in the demonftration.
In order to give a bet-
ter idea of this fyftem, and A
of the advantages gained by its adoption, let us take a right
X INTRODUCTION.
angled triangle, and exprefs fome of its properties both by
colours and the method generally employed.
Some of the properties of the right angled triangle ABC,
expreffed by the method generally employed.
1 . The angle BAC, together with the angles BCA and
ABC are equal to two right angles, or twice the angle ABC.
2. The angle CAB added to the angle ACB will be equal
to the angle ABC.
3. The angle ABC is greater than either of the angles
BAC or BCA.
4. The angle BCA or the angle CAB is lefs than the
angle ABC.
5. If from the angle ABC, there be taken the angle
BAC, the remainder will be equal to the angle ACB.
6. The fquare of AC is equal to the fum of the fquares
of AB and BC.
The fame properties expreffed by colouring the different parts.
That is, the red angle added to the yellow angle added to
the blue angle, equal twice the yellow angle, equal two
right angles.
-^ + A =
Or in words, the red angle added to the blue angle, equal
the yellow angle.
▲
<^H^ CZ JK^ or
The yellow angle is greater than either the red or blue
angle.
INTRODUCTION. xl
iL
4. jl^^ or
Either the red or blue angle is lefs than the yellow angle.
^^^^^ minus ^HL
In other terms, the yellow angle made lefs by the blue angle
equal the red angle.
That is, the fquare of the yellow line is equal to the fum
of the fquares of the blue and red lines.
In oral demonftrations we gain with colours this impor-
tant advantage, the eye and the ear can be addreffed at the
fame moment, fo that for teaching geometry, and other
linear arts and fciences, in clafTes, the fyftem is the beft ever
propofed, this is apparent from the examples juft given.
Whence it is evident that a reference from the text to
the diagram is more rapid and fure, by giving the forms
and colours of the parts, or by naming the parts and their
colours, than naming the parts and letters on the diagram.
Befides the fuperior limplicity, this fyftem is likewife con-
fpicuous for concentration, and wholly excludes the injuri-
ous though prevalent pradlice of allowing the ftudent to
commit the demonftration to memory ; until reafon, and fadl,
and proof only make impreffions on the underftanding.
Again, when ledluring on the principles or properties of
figures, if we mention the colour of the part or parts re-
ferred to, as in faying, the red angle, the blue line, or lines,
&c. the part or parts thus named will be immediately feen
by all in the clafs at the fame inftant ; not fo if we fay the
angle ABC, the triangle PFQ^the figure EGKt, and fo on ;
xii INTRODUCTION.
for the letters mufl be traced one by one before the fludents
arrange in their minds the particular magnitude referred to,
which often occafions confufion and error, as well as lofs of
time. Alfo if the parts which are given as equal, have the
fame colours in any diagram, the mind will not wander
from the objedl before it ; that is, fuch an arrangement pre-
fents an ocular demonftration of the parts to be proved
equal, and the learner retains the data throughout the whole
of the reafoning. But whatever may be the advantages of
the prefent plan, if it be not fubftituted for, it can always
be made a powerful auxiliary to the other methods, for the
purpofe of introdudlion, or of a more fpeedy reminifcence,
or of more permanent retention by the memory.
The experience of all who have formed fyftems to im-
prefs fadts on the underftanding, agree in proving that
coloured reprefentations, as pidlures, cuts, diagrams, &c. are
more eafily hxed in the mind than mere fentences un-
marked by any peculiarity. Curious as it may appear,
poets feem to be aware of this fadl more than mathema-
ticians ; many modern poets allude to this viiible fyftem of
communicating knowledge, one of them has thus expreffed
himfelf :
Sounds which addrefs the ear are loft and die
In one fhort hour, but thefe which ftrilce the eye,
Live long upon the mind, the faithful fight
Engraves the knowledge with a beam of light.
This perhaps may be reckoned the only improvement
which plain geometry has received fince the days of Euclid,
and if there were any geometers of note before that time,
Euclid's fuccefs has quite eclipfed their memory, and even
occalioned all good things of that kind to be alfigned to
him ; like ^Efop among the writers of Fables. It may
alfo be worthy of remark, as tangible diagrams afford the
only medium through which geometry and other linear
INTRODUCTION. xiii
arts and fciences can be taught to the blind, this vifible fys-
tem is no lefs adapted to the exigencies of the deaf and
dumb.
Care muft be taken to fliow that colour has nothing to
do with the lines, angles, or magnitudes, except merely to
name them. A mathematical line, which is length with-
out breadth, cannot poffefs colour, yet the jundtion of two
colours on the fame plane gives a good idea of what is
meant by a mathematical line ; recolledt we are fpeaking
familiarly, fuch a jundlion is to be underftood and not the
colour, when we fay the black line, the red line or lines, &c.
Colours and coloured diagrams may at firfl: appear a
clumiy method to convey proper notions of the properties
and parts of mathematical figures and magnitudes, how-
ever they will be found to afford a means more refined and
extenfive than any that has been hitherto propofed.
We fliall here define a point, a line, and a furface, and
demonflrate a propofition in order to fhow the truth of this
affertion.
A point is that which has pofition, but not magnitude ;
or a point is pofition only, abftradled from the confideration
of length, breadth, and thicknefs. Perhaps the follow-
ing defcription is better calculated to explain the nature of
a mathematical point to thofe who have not acquired the
idea, than the above fpecious definition.
Let three colours meet and cover a
portion of the paper, where they meet
is not blue, nor is it yellow, nor is it
red, as it occupies no portion of the
plane, for if it did, it would belong
to the blue, the red, or the yellow
part; yet it exifts, and has pofition
without magnitude, fo that with a Uttle refledlion, this June-
XIV
INTRODUCTION.
tion of three colours on a plane, gives a good idea of a
mathematical point.
A line is length without breadth. With the afliftance
of colours, nearly in the fame manner as before, an idea of
a line may be thus given : —
Let two colours meet and cover a portion of the paper;
where they meet is not red, nor is it
blue ; therefore the jundlion occu-
pies no portion of the plane, and
therefore it cannot have breadth, but
only length : from which we can
readily form an idea of what is meant by a mathematical
line. For the purpofe of illuftration, one colour differing
from the colour of the paper, or plane upon which it is
drawn, would have been fufficient ; hence in future, if we
fay the red line, the blue line, or lines, &c. it is the junc-
tions with the plane upon which they are drawn are to be
underftood.
Surface is that which has length and breadth without
thicknefs.
When we confider a folid body
(PQ), we perceive at once that it
has three dimenfions, namely : —
length, breadth, and thicknefs ;
fuppofe one part of this folid (PS)
to be red, and the other part (QR)
yellow, and that the colours be
diflinft without commingling, the
blue furface (RS) which feparates
thefe parts, or which is the fame
S thing, that which divides the folid
without lofs of material, mufl be
without thicknefs, and only poffeffcs length and breadth ;
INTRODUCTION.
XV
this plainly appears from reafoning, limilar to that juft em-
ployed in defining, or rather delcribing a point and a line.
The propofition which we have felefted to elucidate the
manner in which the principles are applied, is the fifth of
the firft Book.
In an ifofceles triangle ABC, the
internal angles at the bafe ABC,
ACB are equal, and when the fides
AB, AC are produced, the exter-
nal angles at the bafe BCE, CBD
are allb equal.
Produce _i__ and
make ■■■■ "^
Draw ^— — and
(B. i.pr. 3.)
and
and
common
and
^ = -^ (B. I. pr. 4.)
Again in >^ and N. t ^
xvi INTRODUCTION.
and ^ = ^;
and ^^^ ^ ^^^ (B. i. pr. 4).
But
C^E. D.
By annexing Letters to the Diagratn.
Let the equal fides AB and AC be produced through the
extremities BC, of the third Tide, and in the produced part
BD of either, let any point D be afllimed, and from the
other let AE be cut off equal to AD (B. i. pr. 3). Let
the points E and D, fo taken in the produced fides, be con-
nedted by ftraight lines DC and BE with the alternate ex-
tremities of the third fide of the triangle.
In the triangles DAC and EAB the fides DA and AC
are refpedlively equal to EA and AB, and the included
angle A is common to both triangles. Hence (B i . pr. 4.)
the line DC is equal to BE, the angle ADC to the angle
AEB, and the angle ACD to the angle ABE ; if from
the equal lines AD and AE the equal fides AB and AC
be taken, the remainders BD and CE will be equal. Hence
in the triangles BDC and CEB, the fides BD and DC are
refpedively equal to CE and EB, and the angles D and E
included by thofe fides are alfo equal. Hence (B. i. pr. 4.)
INTRODUCriON. xvii
the angles DBC and ECB, which are thofe included by
the third fide BC and the productions of the equal fides
AB and AC are equal. Alfo the angles DCB and EBC
are equal if thofe equals be taken from the angles DCA
and EBA before proved equal, the remainders, which are
the angles ABC and ACB oppofite to the equal fides, will
be equal.
Therefore in aii ifofceles triangle y &c.
Q^E. D.
Our object in this place being to introduce the fyftem
rather than to teach any particular fet of propofitions, we
have therefore feledled the foregoing out of the regular
courfe. For fchools and other public places of infi:rud:ion,
dyed chalks will anfwer to defcribe diagrams, &c. for private
ufe coloured pencils will be found very convenient.
We are happy to find that the Elements of Mathematics
now forms a confiderable part of every found female edu-
cation, therefore we call the attention of thofe interefiied
or engaged in the education of ladies to this very attractive
mode of communicating knowledge, and to the fucceeding
work for its future developement.
We fhall for the prefent conclude by obferving, as the
fenfes of fight and hearing can be fo forcibly and infiianta-
neously addreffed alike with one thoufand as with one, the
million might be taught geometry and other branches of
mathematics with great eafe, this would advance the pur-
pofe of education more than any thing that might be named,
for it would teach the people how to think, and not what
to think ; it is in this particular the great error of education
originates.
XVlll
THE ELEMENTS OF EUCLID.
BOOK I.
DEFINITIONS.
I.
A point is that which has no parts.
II.
A line is length without breadth.
III.
The extremities of a line are points.
IV.
A ftraight or right line is that which lies evenly between
its extremities.
V.
A furface is that which has length and breadth only.
VI.
The extremities of a furface are lines.
VII.
A plane furface is that which lies evenly between its ex-
tremities.
VIII.
A plane angle is the inclination of two lines to one ano-
ther, in a plane, which meet together, but are not in the
fame diredlion.
IX.
^ A plane redlilinear angle is the inclina-
^r tion of two ftraight lines to one another,
^^^ which meet together, but are not in the
ir fame flraight line.
BOOK I. DEFINITIONS.
XIX
When one ftraight line Handing on ano-
ther ftraight Hne makes the adjacent angles
equal, each of thefe angles is called a rigkf
angle, and each of thefe lines is faid to be
perpendicular to the other.
A
XI.
An obtufe angle is an angle greater
than a right angle.
XII.
An acute angle is an angle lefs than a
right angle.
XIII.
A term or boundary is the extremity of any thing.
XIV.
A figure is a furface enclofed on all fides by a line or lines.
XV.
A circle is a plane figure, bounded
by one continued line, called its cir-
cumference or periphery ; and hav-
ing a certain point within it, from
which all ftraight lines drawn to its
circumference are equal.
XVI.
This point (from which the equal lines are drawn) is
called the centre of the circle.
XX BOOK I. DEFINITIONS.
XVII.
A diameter of a circle is a ftraight line drawn
through the centre, terminated both ways
in the circumference.
XVIII.
A femicircle is the figure contained by the
diameter, and the part of the circle cut off
by the diameter.
XIX.
A fegment of a circle is a figure contained
by a ftraight line, and the part of the cir-
cumference which it cuts off.
^•••••••*
••'•'
XX.
A figure contained by ftraight lines only, is called a redli-
linear figure.
XXI.
A triangle is a redlilinear figure included by three fides.
XXII.
A quadrilateral figure is one which is bounded
by four fides. The fi:raight lines ■^— «— .
and .^_«— i«> connecfting the vertices of the
oppofite angles of a quadrilateral figure, are
called its diagonals.
XXIII.
A polygon is a redilinear figure bounded by more than
four fides.
BOOK I. DEFINITIONS.
XXI
XXIV.
A triangle whofe three fides are equal, is
faid to be equilateral.
XXV.
A triangle which has only two fides equal
is called an ilbfceles triangle.
XXVI. "
A fcalene triangle is one which has no two fides equal.
XXVII.
A right angled triangle is that which
has a right angle.
XXVIII.
An obtufe angled triangle is that which
has an obtufe angle.
XXIX.
An acute angled triangle is that which
has three acute angles.
XXX.
Of four-fided figures, a fquare is that which
has all its fides equal, and all its angles right
angles.
XXXI.
A rhombus is that which has all its fides
equal, but its angles are not right angles.
XXXII.
u
An oblong is that which has all its
angles right angles, but has not all its
fides equal.
xxii BOOK L POS'lVLATES.
XXXIII.
A rhomboid is that which has its op-
pofite fides equal to one another,
but all its fides are not equal, nor its
angles right angles.
XXXIV.
All other quadrilateral figures are called trapeziums.
XXXV,
^^—--^,^g„^^^ Parallel ftraight lines are fuch as are in
^^^^^^^^^^ the fame plane, and which being pro-
duced continually in both directions,
would never meet.
POSTULATES.
I.
Let it be granted that a flraight line may be drawn from
any one point to any other point.
II.
Let it be granted that a finite ftraight line may be pro-
duced to any length in a ftraight line.
III.
Let it be granted that a circle may be defcribed with any
centre at any diflance from that centre.
AXIOMS.
I.
Magnitudes which are equal to the fame are equal to
each other.
II.
If equals be added to equals the fums will be equal.
BOOK I. AXIOMS. xxiii
III.
If equals be taken away from equals the remainders will
be equal.
IV.
If equals be added to unequals the fums will be un-
equal.
V.
If equals be taken away from unequals the remainders
will be unequal.
VI.
The doubles of the fame or equal magnitudes are equal.
VII.
The halves of the fame or equal magnitudes are equal.
VIII.
Magnitudes which coincide with one another, or exactly
fill the fame fpace, are equal.
IX.
The whole is greater than its part,
X.
Two ftraight lines cannot include a fpace.
XI.
All right angles are equal.
XII.
If two ftraight lines ( } meet a third
ftraight line ( ) fo as to make the two interior
angles ( and jj^ ) on the fame fide lefs than
two right angles, thefe two ftraight lines will meet if
they be produced on that fide on which the angles
are lefs than two right angles.
XXIV
BOOK I. ELUCIDATIONS.
The twelfth axiom may be expreffed in any of the fol-
lowing ways :
1 . Two diverging ftraight lines cannot be both parallel
to the fame flraight line.
2. If a ftraight line interfeft one of the two parallel
ftraight lines it mufl alfo interfedt the other.
3. Only one ftraight line can be drawn through a given
point, parallel to a given ftraight line.
Geometry has for its principal objefts the expofition and
explanation of the properties oi figure, and figure is defined
to be the relation which fubfifts between the boundaries of
fpace. Space or magnitude is of three kinds, linear, fuper-
ficial, ■Si.w^foUd.
Angles might properly be confidered as a fourth fpecies
of magnitude. Angular magnitude evidently confifts of
parts, and muft therefore be admitted to be a fpecies ol
quantity The ftudent muft not fuppofe that the magni-
tude of an angle is affefted by the length
of the ftraight lines which include it, and
of whofe mutual divergence it is the mea-
fure. The vertex of an angle is the point
where \}[\& fides or the legs of the angle
meet, as A.
An angle is often defignated by a fingle letter when its
legs are the only lines which meet to-
gether at its vertex. Thus the red and
blue lines form the yellow angle, which
in other fyftems would be called the
angle A. But when more than two
B lines meet in the fame point, it was ne-
ceflary by former methods, in order to
avoid confufion, to employ three letters
to defignate an angle about that point.
BOOK I. ELUCIDATIONS. xxv
the letter which marked the vertex of the angle being
always placed in the middle. Thus the black and red lines
meeting together at C, form the blue angle, and has been
ufually denominated the angle FCD or DCF The lines
FC and CD are the legs of the angle; the point C is its
vertex. In like manner the black angle would be defignated
the angle DCB or BCD. The red and blue angles added
together, or the angle HCF added to FCD, make the angle
HCD ; and fo of other angles.
When the legs of an angle are produced or prolonged
beyond its vertex, the angles made by them on both fides
of the vertex are faid to be vertically oppofite to each other :
Thus the red and yellow angles are faid to be vertically
oppofite angles.
Superpojition is the procefs by which one magnitude may
be conceived to be placed upon another, fo as exadlly to
cover it, or fo that every part of each fhall exadly coin-
cide.
A line is faid to be produced, when it is extended, pro-
longed, or has its length increafed, and the increafe of
length which it receives is called its produced part, or its
produSlion.
The entire length of the line or lines which enclofe a
figure, is called its perimeter. The firft fix books of Euclid
treat of plain figures only. A line drawn from the centre
of a circle to its circumference, is called a radius. The
lines which include a figure are called \isjides. That fide
of a right angled triangle, which is oppofite to the right
angle, is called the hypotenufe. An oblong is defined in the
fecond book, and called a reSlangle. All the lines which
are confidered in the firfl: fix books of the Elements are
fuppofed to be in the fame plane.
The Jiraight-edge and compajfcs are the only inflruments.
xxvi BOOK I. ELUCIDATIONS.
the ufe of which is permitted in Euclid, or plain Geometry.
To declare this reflridlion is the objedl of the pojiulates.
The Axioms of geometry are certain general proportions,
the truth of which is taken to be felf-evident and incapable
of being eftabliflied by demonftration.
Propojitions are thofe refults which are obtained in geo-
metry by a procefs of reafoning. There are two fpecies of
propofitions in geometry, problems and theorems.
A Problem is a propofition in which fomething is pro-
pofed to be done ; as a line to be drawn under fome given
conditions, a circle to be defcribed, fome figure to be con-
rtrudled, &c.
Th.t folution of the problem confifts in fhowing how the
thing required may be done by the aid of the rule or ftraight-
edge and compafTes.
The demonftration confifts in proving that the procefs in-
dicated in the folution really attains the required end.
A Theorem is a propofition in which the truth of fome
principle is afi^erted. This principle mufl: be deduced from
the axioms and definitions, or other truths previously and
independently ellabliihed. To fhow this is the objedl of
demonftration.
A Problem is analogous to a poftulate.
A Theorem refembles an axiom.
A Pojlulate is a problem, the folution of which is afiiimed.
An Axiom is a theorem, the truth of which is granted
without demonftration.
A Corollary is an inference deduced immediately from a
propofition.
A Scholium is a note or obfervation on a propofition not
containing an inference of fufiicient importance to entitle it
to the name of a corollary.
A Lemma is a propofition merely introduced for the pur-
pofe of eftabliftiing fome more important propofition.
xxvu
SYMBOLS AND ABBREVIATIONS.
,*, exprefles the word therefore.
*,' becaufe.
zz equal. This fign of equaHty may
be read equal to, or is equal to, or are equal to ; but
any difcrepancy in regard to the introdudlion of the
auxiliary verbs Is, are, &c. cannot affedl the geometri-
cal rigour.
^ means the fame as if the words ' not equal' were written.
r~ fignifies greater than.
^ . . . . lefs than.
Cjl . . . . not greater than.
j] . . . . not lefs than.
-\- is vtzdplus [fjiore), the fign of addition ; when interpofed
between two or more magnitudes, fignifies their fum.
— is read minus {lefs), fignifies fubtracftion ; and when
placed between two quantities denotes that the latter
is to be taken from the former.
X this fign exprefi"es the produdl of two or more numbers
when placed between them in arithmetic and algebra ;
but in geometry it is generally ufed to exprefs a rect-
angle, when placed between " two flraight lines which
contain one of its right angles." A reBangle may alfo
be reprefented by placing a point between two of its
conterminous fides.
: :; : exprefies an analogy or proportion ; thus, if A, B, C
and D, reprefent four magnitudes, and A has to
B the fame ratio that C has to D, the propofition
is thus briefly written,
A : B ; : C : D,
A : B = C : D,
A C
°'"b = d.
This equality or famenefs of ratio is read,
xxviii STMBOLS AND ABBREVIAnONS.
as A is to B, fo is C to D ;
or A is to B, as C is to D.
II fignifies parallel to.
J_ . . . . perpendicular to.
. angle.
. right angle.
CIS
two right angles,
^1^ or I N briefly defignates a point.
C =, or ^ fignifies greater, equal, or lefs than.
The fquare defcribed on a line is concifely written thus.
In the fame manner twice the fquare of, is expreffed by
2 \
def. fignifies definition.
pos pofiulate.
ax axiom.
hyp hypothefis. It may be necefiary here to re-
mark, that the hypothefis is the condition aflumed or
taken for granted. Thus, the hypothefis of the pro-
pofition given in the Introduction, is that the triangle
is ifofceles, or that its legs are equal.
conft confiruElion. The confiruBion is the change
made in the original figure, by drawing lines, making
angles, defcribing circles, &c. in order to adapt it to
the argument of the demonfi:ration or the folution of
the problem. The conditions under which thefe
changes are made, are as indisputable as thofe con-
tained in the hypothefis. For infi:ance, if we make
an angle equal to a given angle, thefe two angles are
equal by conftrudlion.
Q^E. D ^lod erat detnonfirandum.
Which was to be demonftrated.
CORRIGENDA. xxix
Faults to be correEied before reading this Volu7Jie.
Page 13, line 9, /or def. 7 read ^z.L 10.
45, laft line, /or pr. 19 r^^^ pr. 29.
54, line 4 from the bottom, /or black and red line read blue
and red line.
59, line 4, /or add black line fquared read add blue line
fquared.
60, line 17, /or red line multiplied by red and yellow line
read red line multiplied by red, blue, and yellow line.
76, line 11, for def. 7 read dt?. 10.
81, line lOyfor take black line r^i2ii take blue line.
105, line 11, for yellow black angle add blue angle equal red
angle read yellow black angle add blue angle add red
angle.
129, laft line, /or circle read triangle.
141, line I, /or Draw black line read Draw blue line.
196, line 3, before the yellow magnitude infert M.
(Euclib.
BOOK I.
PROPOSITION I. PROBLEM.
N a given finite
firaight line ( )
to dejcribe an equila-
teral triangle.
Defcribe I "^^ and
o
(postulate 3.); draw and — (poft. i.).
then will \ be equilateral.
(def. 15.);
— (def. 15.),
• ^_ -mm
— (axiom, i .) ;
and therefore \^ is the equilateral triangle required.
Q^E. D
B
BOOK I. PROP. II. PROB.
ROM aghenp'jhit ( ■■ ),
to draic ajiraight line equ.al
to a green finite firaight
line ( ).
Draw — — — — (poil. I.), defcribe
Afpr. I.), produce — — (poll.
o
2.), defcribe
(poft. 3.), and
(poll. 3.) ; produce — ^— "" (port. 2.), ther
is the line required.
For
and
(def. 15.),
(conll.), .*.
(ax. 3.), but (def. 15.'
drawn from the given point (
is equal the given line
Q. E. D.
BOOK I. PROP. in. PROP.
ROM the greater
( "—) of
tivo given Jiraight
lines, to cut off a part equal to
the kfs ( ).
Draw
(poll:. 3 .), then
(pr. 2.) ; defcribe
For
and
(def. 15.),
(conll.) ;
(ax. I.).
Q. E. D.
BOOK I. PROP. IF. THEOR.
F two triangles
have two fides
of the one
refpeSlively
equal to two fdes of the
other, ( I to '
and ^__ to w^^m. ) and
the angles { and ^ )
contained by thofe equal
fdes alfo equal ; then their bafes or their fdes (-^-^— and
^^^^) are alfo equal : and the remaining and their remain-
ing angles oppofte to equal fides are refpeSlively equal
( ^^ =: ^^ and ^^ n ^^ ) ; and the triangles are
equal in every refpeB.
Let the two triangles be conceived, to be fo placed, that
the vertex of the one of the equal angles.
or
fliall fall upon that of the other, and
with
then will
^^— to coincide
coincide with » i if ap-
plied: confequently
will coincide with
or two flraight lines will enclofe a fpace, which is impoffible
(ax. lo), therefore
and
^=»
^ ^^ , and as the triangles
* = >
A-^
coincide, when applied, they are equal in every refpedl.
Q. E. D.
BOOK I. PROP. V. THEOR.
N anj ifofceles triangle
A
if the equal Jides
be produced, the external
angles at the bafe are equal, and the
internal angles at the bafe are alfo
equal.
Produce
and
y (poft. 2.), take
— - = 9 (pr- 3-);
draw -i^— — » and n .
Then in
both, and
A A
/ \ and / \ we have,
= (conft.), A
common to
(hyp.) /. Jk =
and ^ = ^ (pr. 4.).
^ = ^ and
1^^ -zz ^^ \ and ^^» ^ ^^ (pr. 4.) but
^ = ^ "*' Jk = JL ^'-^'
Q. E. D.
BOOK I. PROP. Ft. THEOR.
A
and
N any triangle ( / \ ) ;/'
two angles ( ' and ^L )
are equal, the Jides ( ■— ■
■~ ) oppofite to them are alfo
equal.
For if the fides be not equal, let one
of them I — ■ be greater than the
other
and from it cut off
(pr. 3.), draw-
Then
(conft.)
m
A.naA,
(hyp.)
anc
common,
,*. the triangles are equal (pr. 4.) a part equal to the whole,
which is abfurd ; ,*, neither of the fides — "» or
' is greater than the other, /. hence they are
equal
Q^E. D.
BOOK I. PROP. FII. THEOR.
N the fame bafe (■
■), a7id on
the fa}7ie Jide of it there cannot be tivo
triangles having their conterminous
fides ( and — ^— ^
•— — ■ and «i^i— ii^—) at both extremities of
the bafe, equal to each other.
When two triangles ftand on the fame bale,
and on the fame iide of it, the vertex of the one
Ihall either fall outlide of the other triangle, or
within it ; or, laftly, on one of its lides.
llructed fo that
#=''
If it be poffible let the two triangles be con-
'«■ rzzzz — zizzz f ^^^"
draw ——---- and,
= ^ (Pr- 5-)
.'. ^^ ^ ^^ and
but (pr. 5.) yf = ^^
therefore the two triangles cannot have their conterminous
which is abfurd.
fides equal at both extremities of the bafe.
Q. E. D.
BOOK I. PROP. Fill. THEOR.
F two triangles
have two Jides
of the one refpec-
tjvely equal to
two Jides of the other
and .—m^ =r ),
and alfo their bafes (
^ •— ), equal ; then the
and
)
angles (
contained by their equal Jides
are alfo equal.
If the equal bafes
and
be conceived
to be placed one upon the other, fo that the triangles fhall
lie at the fame fide of them, and that the equal fides
«______ and .i.....i_ , —«-.—. and _____ be con-
terminous, the vertex of the one mufi: fall on the vertex
of the other ; for to fuppofe them not coincident would
contradidl the laft propofition.
Therefore the fides
cident with
and .
, and
., being coin-
k-k
Q. E. D.
BOOK I. PROP. IX. PROP.
Take
O bifeB a given reSlilinear
angle {^ J.
(PJ*- 3-)
draw
, upon which
defcribe ^^ (pr. i.).
draw
Becaufe _ = ..^... (confl.)
and ^^^— common to the two triangles
and
(conft.).
4
= (pr. 8.)
Q. E. D.
10
BOOK I. PROP. X. PROB.
O i>tye^ a given finite Jlraight
line [f^^^mmmmwm'^.
and
common to the two triangles.
Therefore the given line is bifefted.
Q;E. D.
BOOK L PROP. XL PROB.
II
( :
a perpendicular.
ROM a given
point ( I ),
in a given
Jlraight line
— ), to draw
Take any point (•
cut off
) in the given line,
(pr- 3-)'
/ \ (Pr. I.),
conftrudl
draw — — and it fliall be perpendicular to
the given line.
For
(conft.)
(conft.)
and
- common to the two triangles.
Therefore ^|| z:z.
J.
(pr. 8.)
(def. 10.).
C^E.D.
12
BOOK I. PROP, XII. PROB.
O draw a
Jlraight line
perpendicular
to a given
/ indefinite Jlraight line
(^^^ ^ from a given
[point /ys. ) "without.
With the given point /|\ as centre, at one fide of the
line, and any diftance — ^^— capable of extending to
the other fide, defcribe
Make
draw ^
(pr. 10.)
and
then
For (pr. 8.) lince
(conft.)
and
common to both,
= (def. 15.)
and
(def. 10.).
Q. E. D.
BOOK I. PROP. XIII. THEOR.
13
HEN a Jlralght line
( ..m^^m^ ) Jlanding
upon another Jlraight
line ( )
makes angles with it; they are
either two right angles or together
equal to two right angles.
If
be _L to
gf..A=C£^
then,
(def. 7.).
But if
draw
be not _L to — — —
J. ;(pr. II.)
(conft.).
Q. E. D.
H
BOOK I. PROP. XIV. THEOR.
F two Jiraight lines
fneeting a thirdjlraight
line (i ' ), at the
fame pointy and at oppofite Jides of
it, make with it adjacent angles
and
A
) egual to
two right angles ; thefe fraight
lines lie in one continuous Jiraight
line.
For, if pofTible let
and not
be the continuation of
then
+
but by the hypothefis
4 = ^
+
(ax. 3.) ; which is abfurd (ax. 9.).
, is not the continuation of
and
the like may be demonftrated of any other flraight line
except , ,*, ^-^— is the continuation
of
Q. E. D.
BOOK I. PROP. XV. THEOR.
15
gles
and
F two right lines (
and ■' ' I ) interfe£t one
another, the vertical an-
and
^
are
equal.
► -
<*
► 4
In the fame manner it may be fliown that
Q^E. D.
i6
BOOK I. PROP. XVI. THEOR.
F a fide of a
is produced, the external
trian-
greater than either of the
internal remote angles
(
▲ .A
)•
Make
Draw
— (pr. lo.).
- and produce it until
■^^— ; draw — ^— • ,
In
and #•••'
► 4
and
(conft. pr. 15.), /. ^m = ^L (pr. 4.),
...f^.A.
In like manner it can be fhown, that if •—-■••
be produced, ^^^^ Q ^^k , and therefore
is [= ^ii.
Q. E. D.
which is ^z
BOOK I. PROP. XVII. THEOR.
17
NY tivo angles of a tri-
angle f * are to-
gether lefs than two right angles.
Produce
+
then will
^Oi
But, mik [= Mk (pr- 16.)
and in the fame manner it may be Ihown that any other
two angles of the triangle taken together are lefs than two
right angles.
Q;E. D.
i8
BOOK I. PROP. XVIIL THEOR.
A
N any triangle
if one Jide vbm* be
greater than another
•^^mmmm-^ ^ the aUgk Of-
pojite to the greater Jide is greater
than the angle oppoftte to the lefs.
1. e.
^
Make
Then will
(pr. 3.), draw
A.A
(pr- 5-) J
but
i£k
(pr. 16.)
and much more
IS
^->
Q. E. D.
BOOK I. PROP. XIX. THEOR.
19
A
F m any triangle
one angle J/j^ be greater
than another ^^^ the Jide
which is oppojite to the greater
angle, is greater than the Jide
oppojite the lefs.
If
be not greater than
or
then muft
If
then
which is contrary to the hypothefis.
— is not lefs than •^■— ^—j for if it were,
which is contrary to the hypothefis :
Q. E. D.
20
BOOK I. PROP. XX. THEOR.
NY two fides
and iBMMH
of a
triangle
Z\
taken together are greater than the
third fide ( ).
Produce
and
make ><
(pr- 3-);
draw
Then becaufe ------ ^
(conft.).
(ax. 9.)
+
and ,*,
+
(pr. 19.)
Q.E.D
BOOK I. PROP. XXL THEOR.
21
F from any point ( / )
within a triangle
' Jlraight lines be
drawn to the extremities of one fide
( ), thefe lines mujl he toge-
ther lefs than the other two fdes, but
muJl contain a greater angle.
Produce
+
add
to each.
(pr. 20.),
+
+
(ax. 4.)
In the fame manner it may be fhown that
... + [Z +
which was to be proved.
4
■.A
(pr. 16.),
(pr. 16.),
Q^E.D.
22
BOOK I. PROP. XXII. THEOR.
\IVE'N three ng/it
lines < -■••—
the fum of any
two greater than
the third, to conJlru6i a tri-
angle whofe Jides Jhall be re-
fpeSlively equal to the given
lines.
■■■•■«a««^«M
AfTume
Draw — — ^
and -^— • s:
With
defcribe
and
and
0
I (pr. 2.).
as radii,
(poft. 3.);
draw and
then will
For
and ■
be the triangle required.
"' i
Q. E. D.
BOOK I. PROP. XXIII. PROB. 23
iT a given point ( ) in a
given Jiraight line (^^^»»— ■),
to make an angle equal to a
given re 51 i lineal angle (.^^j^ )•
Draw — — — . between any two points
in the legs of the given angle.
Conftruct v (pr. 22.)
fo that — ^^^ = .
and
Then jgj^ = ^J^ (pr. 8.).
Q. E. D.
24
BOOK I. PROP. XXir. THEOR.
X>
F two triangles
have two fides of
the one refpec-
tively equal to
twofdes of the other (
to and ------
to ), and if one of
A
the angles ( <3. .\ ) contain-
ed by the equal fdes be
greater than the other (c.»«^), the fide ( ^-^-^^ ) isohich is
oppofte to the greater angle is greater than thefde ( - . . . )
which is oppofte to the lefs angle.
Make
and —
L^ - ly (pr. 23.),
= (pr- 3-).
draw ..-••-■-•» and -■——■.
Becaufe ^— — ^ 3: — •— — (ax. i. hyp. conft.)
but
and .*.
^ = ^ (F-
but
(pr. 19.)
(pr.4.)
Q. E. D.
BOOK I. PROP. XXV. THEOR.
25
F two triangles
have two fides
(" '■"■' and
) of the
one refpeBively equal to two
fides ( and — — )
of the other, but their bafes
unequal, the angle fubtended
by the greater bafe (««—■—■)
of the one, muji be greater
than the angle fubtended by
the lefs bafe ("■"■■"*•) of the other.
^Im- ^ , C or H] ^^ ^^^ is not equal to ^^
^^ •=. ^^ then ^^^^ := — — i- (pr. 4.)
for if
which is contrary to the hypothefis ;
^H^ is not lefs than ^^
for if A :ti A
then i "H ' (pr. 24.),
which is alfo contrary to the hypothefis :
/.A [= A.
Q^E. D.
26 BOOK I. PROP. XXVL THEOR.
Case I.
F two triangles
have two angles
of the one re-
fpedlively equal
to two angles of the other.
(
and
Case II.
tf)
Let
y), and a fide
of the one equal to afde of
the other fmilarly placed
with refpeSl to the equal
angles, the remaining fdes
and angles are refpeSlively
equal to one another.
CASE I.
and I which lie between
the equal angles be equal,
then -^— — ^ ^^— ■•••
For if it be poflible, let one of them -i
greater than the other ;
be
In X \ and X ^
we have
M = A
(pr.4.)
BOOK I. PROP. XXVI. THEOR. 27
but A = iH (hyp.)
and therefore g^^ =: ^|B, which is abfurd ;
hence neither of the fides — ^— ■— and — ■^■■■- is
greater than the other; and .*. they are equal;
and 4 = 4,
(pr. 4.).
CASE II.
Again, let ^— — • ^ ■— — — ^ which lie oppofite
the equal angles flik and ^^^ . If it be poflible, let
Then in ' ^ and J^^^ we have
= and /^ = J^,
I'ut H^ = JBi^ (hyp.)
.*. jf^ = ^^^ which is abfurd (pr. 16.).
Confequently, neither of the fides ^"i— i"«» or ^-^"i—^ is
greater than the other, hence they muft be equal. It
follows (by pr. 4.) that the triangles are equal in all
refpedls.
Q^E. D.
28
BOOK I. PROP. XXVII. THEOR.
F ajlralght line
( ) meet-
i?2g tivo other
Jiraight lines,
- and ) makes
with them the alternate
angles (
and
) equal, thefe two Jiraight lines
are parallel.
If
be not parallel to
they fliall meet
when produced.
If it be poflible, let thofe lines be not parallel, but meet
when produced ; then the external angle ^^ is greater
than flHik>^ (pr. i6),but they are alfo equal (hyp.), which
is abfurd : in the fame manner it may be ihown that they
cannot meet on the other fide ; ,*, they are parallel.
Q. E. D.
BOOK I. PROP. XXFIIL THEOR.
29
(-
F ajlraight line
ting two other
Jlraight lines
makes the external equal to
the internal and oppojite
angle, at the fame Jide of
the cutting line {namely.
yl, or if it makes the two internal angles
at the fame ftde ( ^l^ and ^F , or f/^ and ^^^)
together equal to two right angles, thofe two Jlraight lines
are parallel.
Firft, if
1^ =^^ , then Jjj^ = ^r (pr. i
mL = W /. II (pr. 27.).
Secondly, if
then
+
(pr. 13.),
(ax. 3.)
(pr. 27.)
C^E. D.
30
BOOK I. PROP. XXIX. THEOR.
STRAIGHT /ine
( ) f^^^i'"g on
two parallel Jiraight
» lines ( ■mmmim^mm and
•), makes the alternate
angles equal to one another ; and
alfo tlie external equal to tlie in-
ternal and oppojite angle on the
fame Jide ; and the two internal
angles on the fa?ne Jide together
equal to two right angles.
For if the alternate angles
and
▲
be not equal,
draw
», making
A
Therefore
(pr- 23)-
(pr. 27.) and there-
fore two ftraight lines which interfed: are parallel to the
fame flraight line, which is impoflible (ax. 1 2).
Hence the alternate angles ^^ and ^|^ are not
unequal, that is, they are equal: =: ^^^ (pr. 15);
.*. jl^ = l/^ , the external angle equal to the inter-
nal and oppofite on the fame iide : if ^^W be added to
both, then
A
+
i
^CLi
(pr. 13)-
That is to fay, the two internal angles at the fame fide of
the cutting line are equal to two right angles.
Q. E. D.
BOOK I. PROP. XXX. THEOR.
3^
TRAIGHT /mes ( _Z)
lohich are parallel to the
fame Jlratght line ( ),
are parallel to one another.
Let
interfedl
Then,
= ^^ = iJB (pr. 29.),
(pr. 27.)
Q. E. D.
32 BOOK I. PROP. XXXI. PROB.
ROM a given
point /^ to
draw ajiraight
line parallel to a given
Jlraight line (——•).
Draw
from the point / to any point
in
make
then —
(pr. 23.),
- (pr. 27.).
Q. E. D.
4
BOOK I. PROP. XXXII. THEOR.
33
F any fide (-
•)
of a triangle be pro-
duced, the external
^figl^ ( ^^^) '-^ ^qual
to the fum of the two internal and
oppofte angles ( aiid ^^^ ) ,
and the three internal angles of
every triangle taken together are
equal to two right angles.
Through the point / draw
II (pr. 3i-)-
Then
(pr. 29.),
and therefore
(pr. 13.).
J
-dy
Q. E. D.
34
BOOK I. PROP. XXXIII. THEOR.
TRAIGHT fines (-
and ) which join
the adjacent extremities of
two equal and parallel Jiraight
~— — and "•»..---=. ), are
themf elves equal and parallel.
Draw
the diagonal.
(hyp.)
and
(pr. 29.)
common to the two triangles ;
■, and
▼ = 4
(pr. 4.) ;
and /.
(pr. 27.).
Q. E. D.
BOOK I. PROP. XXXIV. THEOR.
35
HE ofpofite Jides and angles of
any parallelogram are equal,
and the diagonal (i^— ^^— )
divides it into two equal parts.
Since
= A
^ = t
(pr. 29.)
and
common to the two triangles.
/. \
\ (pr- 26.)
and ^^W = ^^M (^^'^ '
Therefore the oppofite fides and angles of the parallelo-
gram are equal : and as the triangles
.N.""^
are equal in every refpect (pr. 4,), the diagonal divides
the parallelogram into two equal parts.
Q. E. D.
36 BOOK I. PROP. XXXV. THEOR.
ARALLELOGRAMS
on the fame bafe, and
between the fame paral-
lels, are {in area) equal.
and
But,
On account of the parallels,
_Kpr. 29.)
(Pi-- 34-)
(pr. 8.)
r=?
minus
minus
r=
Q^E. D.
BOOK I. PROP. XXXVI. THEOR.
37
ARALLELO-
GRAMS
1
is*
( ^^ and ) on
equal bafes, and between the
fame parallels, are equal.
Draw
and ---..-— ,
■, by (pr. 34, and hyp.);
= and II "— (pr. 33.)
And therefore
but
J
1.1
is a parallelogram :
(pr- 35-)
(ax. I.).
Q. E. D.
38 BOOK I. PROP. XXXFII. THEOR.
RIANGLES
on the fame bafe (•
■)
and between the fame paral-
lels are equal.
Draw
Produce
\ fpr. ^i
(pr- 3I-)
1—M. and ^^
are parallelograms
on the fame bafe, and between the fame parallels,
and therefore equal, (pr. 35.)
T
=: twice
f
^ twice
4
(■ (pr- 34-)
k.i
Q. E D.
BOOK I. PROP. XXXVIII. THEOR. 39
RIANGLES
;4H ^'ij JH
(^Hi tind jm^ ) on
equal bajes and between •■•
the fame parallels are equal.
Draw
and
II
(pr. 31.).
I #
(pr. 36.);
and
■ i
= twice ^^k
^^ = twice ^H
(pr- 34-)'
i k
(ax. 7.).
Q^E. D.
40
BOOK I. PROP. XXXIX. THEOR,
QUAL triangles
\
and "^ on the fame bafe
( ) and on the fame fide of it, are
between the fvne parallels.
If-^— ■», which joins the vertices
of the triangles, be not || ,
draw II (pr.3i-).
meeting
Draw
Becaufe
(conft.)
but
W.4
(pr- 37-) ■•
(hyp.) ;
A=4
, a part equal to the whole,
which is abfurd.
Ji. ^i^-^-^ ; and in the fame
manner it can be demonflrated, that no other line except
is II ; .-. II .
Q. E. D.
BOOK I. PROP. XL. THEOR.
41
QUAL trian-
gles
(
and M.
)
on equal bafes, and on the
fame Jide, are between the
fame parallels.
If ■ which joins the vertices of triangles
be not II - ,
draw — — — . II — -~—
(pr. 31.),
meeting
Draw
Becaufe
(conft.)
. ^^^- ^ 1^^^ , a part equal to the whole,
which is abfurd.
' 41" ~^^^"^ • ^"f^ in the fame manner it
can be demonftrated, that no other line except
— is II : .-.
Q^E. D.
42
BOOK I. PROP. XLI. THEOR.
Draw
Then
F a paral-
lelogram
A
and a triangle ^^^ are upon
the fame bafe — ^^^ and be tine en
the fame parallels -.—---- and
■ , the parallelogram is double
the triangle.
the diagonal ;
V=J
zz twice
(pr- 37-)
(pr- 34-)
^^ 4
^1^. ^ twice ^H^ .
Q. E. D.
BOOK I. PROP. XLII. THEOR. 43
O conJiruSl a
parallelogram
equal to a given
4
triangle ^^/^andhaV"
ing an angle equal to a given
rectilinear angle ,
Make — — ^ = ■— « (pr. 10.)
Draw ,
Draw I" [j ~'| (pr. 31.)
^1^ := twice y
(pr. 41.)
but ^ z= lA (pr. 38.)
4
Q. E. D.
44 BOOK I. PROP. XLIII. THEOR.
HE complements
and ^^^ cf
the parallelograms ivhicli are about
the diagonal of a parallelogram are
equal.
(pr- 34-)
4. ^^
and JBL = ^
(pi-- 34-)
(ax. 3.)
Q. E. D.
BOOK I. PROP. XLIV. PROB.
45
O a given
Jlraight line
ply a parallelo-
gram equal to a given tri-
angle ( ^^^^' ), and
having an angle equal to
a given reSiilinear angle
( )■
g
wi
th
▲
= ._i
Make
(pr. 42.)
and having one of its fides -— — - conterminous
with and in continuation of 1 m .
Produce w^^mmm^ till it meets ' '■"■' || »»»■«»■
draw prnHnrp it fill if mpptg •■»■-,• continued ;
draw •••««-.• II — «■■ meeting
produced, and produce >•»■•»«
but
(pr. 430
(conft.)
▲ = ▼=▲
(pr. 19. and confl.)
Q. E. D.
BOOK I. PROP. XLF. PROP.
O conjlrudl a parallelogram equal
to a given reSlilinear figure
(
►
) and having an
angle equal to a given reSlilinear angle
Draw
and
K.(t^m
dividing
to
the redtilinear figure into triangles.
Conftrudl
having .„
— apply
(pr.42.)
having
to
having
(pr. 44.)
apply M =
(pr. 44.)
#=►
##= >,
and
Mf mg is a parallelogram, (prs. 29, 14, 30.)
having
Q. E. D.
BOOK I. PROP. XLVI. PROB.
47
PON a given Jlraight line
(— ^^^) to conJlruB a
fquare.
Draw
Draw •
ing .
and
(pr. 1 1, and 3.)
II
drawn ||
>, and meet-
In
^
(conft.)
=: a right angle (conft.)
^H = Hp = ^ "g'^^ ^"gle (pr. 29.),
and the remaining fides and angles muft
be equal, (pr. 34.)
and ,*,
is a fquare. (def. 27.)
Q. E. D.
48 BOOK I. PROP. XLVII. THEOR.
N a right angled triangle
the fquare on the
liypotenufe <• •< is equal to
the fum of the fquares ofthejides, (■
and ).
On
and
defcribe fquares, (pr. 46.)
Draw -.—I
alfo draw
- (pr. 31-)
and
To each add
T
and
Again, becaufe
BOOK I. PROP. XLVII. THEOR.
49
and
twice
= twice ^H •
In the fame manner it may be fhown
that ^^ ^
hence
##
Q E. D.
H
so
BOOK I. PROP. XLVIIL THEOR.
/
F t/ie fquare
of one Jide
{ \ ) f
a triangle is
equal to the fquares of the
other tivo fides (nn.i i
and ), the angle
(
)fubtended by that
fide is a right angle.
Draw ■-
and ^
(prs.11.3.)
and draw —»-«--— alfo.
Since
(conft.)
... "- +
but ^ + -
and — ^— i^- -|-
+
(pr. 47-).
- (hyp.)
and ,*,
confequently
(pr. 8.),
is a right angle.
Q. E. D.
51
BOOK II.
DEFINITION I.
RECTANGLE or a
right angled parallelo-
gram is faid to be con-
tained by any two of its adjacent
or conterminous fides.
Thus : the right angled parallelogram HH[
be contained by the fides — — — ^ and —
or it may be briefly defignated by
is faid to
If the adjacent fides are eq^ual ; i. e. -— — — ^ ^
then — i^»^-« . - which is the expreflion
for the redtangle under
is a fquare, and
is equal to J
and
- or
- or
52
BOOK II. DEFINITIONS.
DEFINITION II.
N a parallelogram,
the figure compokd
of one ot the paral-
lelograms about the diagonal,
together with the two comple-
ments, is called a Gnomon.
Thus
and
are
called Gnomons.
BOOK II. PROP. I. PROP.
53
HE 7-e£langle contained
by two ftraight lines,
one of which is divided
into any number of parts.
= <;+ —
/; equal to the fum of the reBangks
contained by the undivided line, and the fever al parts of the
divided line.
I — — J— — i;
Draw
_L —— — and r=
(prs.2.3.B.i.);
complete the parallelograms, that is to fay,
Draw
\ (pr. 31- B.I.)
L
I
+
- +
Q. E. D.
54
BOOK II. PROP. II. THEOR.
I
I
F a Jlraight line be divided
into any tivo parts ' i ,
the fquare of the -whole line
is equal to the fum of the
reSlangles contained by the whole line and
each of its parts.
-f
I
Defcribe ■■-^^ (B. i. pr. 46.)
Draw — parallel to ----- (B. i. pr. 31 )
I
+
Q. E. D.
BOOK 11. PROP. III. THEOR.
55
F a Jiraig/it line be di-
vided into any two parts
■ 11 ' , the reBangle
contained by the "whole
line and either of its parts, is equal to
the fquare of that part, together with
the reSf angle under the parts.
m
i
= — ^ +
or.
Defcribe
Complete
I
(pr. 46, B. I.)
(pr. 31, B. I.)
Then
+
, but
and
In a fimilar manner it may be readily fhown
that — . — zr m^'i _^ ——. — .
Q. E. D
56
BOOK II. PROP. IF. THEOR.
F a Jiraight line be divided
into any tico parts ,
the fquare of the ii'hole line
is equal to the fquare s of the
parts, together ii-ith twice the reef angle
contained by the parts.
+
+
twice
Defcribe
draw -
and
4-
vpr. 46, B. 1.)
■ port. I.).
(pr. 31, B. I.)
4.4
(pr. 5, B. I.),
(pr. 29, B. I.)
4
500a: //. PROP. IF. THEOR. 57
B
/. by (prs.6,29, 34. B. I.) t,^J is a fquarc ^ — i
For the fame reafons r I is a Iquare := ~"",
« ""~ (pr, 43, b. I.)
I
b"t E— i = C-J+ — +— +
B.
twice >' • ■— ,
Q. E. D.
58
BOOK 11. PROP. V. PROP.
F a Jlraight
line be divided
into two equal
parts and alfo ^
into two unequal parts,
the reSlangle contained by
the unequal parts, together with the fquare of the line between
the points of fe 51 ion, is equal to the fquare of half that line
+
Defcribe IIHIH (pr. 46, B. i.), draw
^ — II — --
and
)
II
(pr.3i,B.i.)
(p. 36, B. I.)
■ - H (p. 43. B. I.)
(ax. 2.)
I-
BOOK II. PROP. r. THEOR.
59
but
and
- (cor. pr. 4. B. 2.)
(conft.)
/. (ax. 2.)
ifl.F-
+
Q. E. D.
6o
BOOK II. PROP. VI. THEOR.
F a Jlraight line be
bifeSled ■
and produced to any
point —^wmmmt ,
the reSlangle contained by the
whole line fo increafed, and the
part produced, together with the
fquare of half the line, is equal
to the fquare of the line made up
of the half, and the produced part .
Defcribe
(pr. 46, B. I.), draw
II
and
(pr. 3i,B. 1.)
(prs. 36, 43, B. I )
but ^H =
(cor. 4, B. 2.)
+
(conft.ax.2.)
Q. E. D.
BOOK 11. PROP. VII. THEOR.
F a Jlraight line be divided
into any two parts wbmw^— ,
the fq liar es of the whole line
and one of the farts are
equal to twice the rectangle contained by
the whole line and that part, together
•with the fquare of the other parts.
6i
Defcribe
Draw -
and
■ ■^■■■■«
, (pr. 46, B. I.)-
(poft. I.),
(pr. 31, B. !.)•
— I (pr- 43. -B. I.),
add ■ = ■-' to both, (cor. 4, B- 2.)
I
(cor. 4, B. 2.)
I
+ ■ +
+
■' + — ^ =
+
Q. E. D.
62
BOOK II. PROP. VIII. THEOR.
E3
F ajlraight line be divided
Into any two parts
, the fquare of
thefum of the whole line
and any one of Its parts. Is equal to
four times the reSlangle contained by
the whole line, and that part together
with the fquare of the other part.
— +
Produce
and make
Conftrudl
draw
J (pr. 46, B. 1.);
(pr. 31, B. I.)
but ^ +
(pr. 4, B. II.)
-^ z= 2. —
(pr. 7, B. II .)
•-+ — ^
+ °-'
Q. E. D.
BOOK 11. PROP. IX. THEOR.
F a Jlraight
line be divided
into two equal
parts ^— — ,.j y
63
and alfo into two unequal
parts ^mmm^'^^m—
^ the
fquares of the unequal
parts are together double
the fquares of half the line,
and of the part between the points offedlion.
^ + ^= 2 ^ + 2
Make — ■ _L and r= —
Draw "..-.—«— and
— II ,— II
or
and draw
= 4
4. = ^
(pr. 5, B.I.) ^ half a right angle,
(cor. pr. 32, B. i.)
(pr. 5, B. I.) =: half a right angle,
(cor. pr. 32, B. i.)
^ a right angle.
4^
lence
(prs. 5, 29, B. I.).
wmmimtm^m^ ■■■■■*
(prs. 6, 34, B. I.)
+
^or +
I
I ■
+
\
(pr. 47, B. I.)
+ 2
Q. E. D.
64
BOOK II. PROP. X. THEOR.
F a Jlraight line
■ be bi-
feBed and pro-
duced to any point
• — , thefquaresofthe
•whole produced line, and of
the produced part, are toge-
ther double of the fquares of
the half litie, and of the line
made up of the half and pro-
duced part.
+
+ ^
Make
and
■— J_ and =1 to
draw ^MvatMit and
or
- f
(pr. 31, B. I.);
draw
alfo.
4
(pr. 5, B. I.) = half a right angle,
(cor. pr. 32, B. i .)
(pr. 5, B. I.) = half a right angle
(cor. pr. 32, B. i.)
4.
m a right angle.
BOOK II. PROP. X. THEOR. 6^
half a right angle (prs. 5, 32, 29, 34, B. i.),
and
.-.., (prs. 6, 34, B. I.). Hence by (pr. 47, B. i.)
Q. E. D.
66
BOOK II. PROP. XI. PROP.
O divide a given fir aight line -^^■■»
in fuch a manner, that the reB angle
contained by the whole line and one
of its parts may be equal to the
fquare of the other.
Defcribe
make «««■
1 1 ■ • «*»■ a
n
draw
take
on
defcribe
(pr. 46, B, I.),
- (pr. 10, B. I.),
(pr. 3, B. I.),
(pr. 46, B. I.),
Produce
— (poft. 2.).
Then, (pr. 6, B. 2.)
2 _ i
+
• •■■••■■
■"■', or,
I
Q^E. D.
BOOK II. PROP. XII THEOR.
67
N any obtufe angled
triangle, thefquare
of the fide fubtend-
ing the obtufe angle
exceeds the fiim of the fquares
of the fides containing the ob-
tufe angle, by twice the rec-
tangle contained by either of
thefe fides andthe produced parts
of the fa?ne from the obtufe
angle to the perpendicular let
fall on it from the oppofite acute
angle.
+
'' by
^ +
2 •
+
By pr. 4, B. 2.
^ + > + 2
add — — — ^ to both
2 _ V
(pr. 47, B. I.)
+
+
• or
■ ; hence '
by 2
'^ (pr. 47, B. I.). Therefore,
^ • ' -"■ + ' +
Q. E. D.
68
BOOK II. PROP. XIII. THEOR.
FIRST.
SECOND.
^m
F^
p^
Br^/^
^
N any tri-
angle, the
fquareofthe
Jidefubtend-
ing an acute angle, is
lefs than the fum of the
fquares of the Jides con-
taining that angle, by twice the reSlangle contained by either
of thefe fides, and the part of it intercepted between the foot of
the perpendicular let fall on it from the oppofte angle, and the
angular point of the acute angle.
FIRST.
+ ■ * by 2
SECOND.
.' -I *by 2
+
2 •
Firft, fuppofe the perpendicular to fall within the
triangle, then (pr. 7, B. 2.)
^■■■> ° -|- ^^^— ^ ^ 2 • ^^i^^"»« • — — -^ ■■■•
add to each ^ihi^'^ then,
I..... "■ -|- _• ^4- - = 2 • ■— • -
+ ' + «
/. (pr- 47. B. I.)
+
BOOK 11. PROP. XIII. THEOR. 69
and .*. ^ Z] ^— "— - + — - by
2 • -■" • ■■■-■i™ .
Next fuppofe the perpendicular to fall without the
triangle, then (pr. 7, B. 2.)
add to each — ■— ■ - then
+ ^ + 2 ... (pr. 47, B. I.),
■J 1 <2 ^_ „ I a
1^— -|- -^.— ^ 2 • ^mM»» . _l-_ -J- ',
Q. E. D.
7°
BOO A' //. PROP. XIV. PROB.
O draw a right line of
•which the fquare flmll be
equal to a given reSli-
linear figure .
fuch that.
Make ^^^^H = ^^V (pr. 45, B. i.),
produce "•- until — — -■. := •
take -■■■.«—- ^ i^— — (pr. 10, B. i.),
Defcribe f \ (poft. 3.),
and produce -^^— to meet it : draw — — ^— ,
(pr. 5, B. 2.),
but — ■ = ' ' " + — "— -(pr. 47, B. I.);
• wmmm^t" ^I ■■■■■■ • «■»« , and
Q. E. D.
BOOK III.
DEFINITIONS.
I.
QUAL circles are thofe whofe diameters are
equal.
II.
A right line is said to touch a circle
when it meets the circle, and being
produced does not cut it.
III.
Circles are faid to touch one an-
other which meet but do not cut
one another.
IV.
Right lines are faid to be equally
diftant from the centre of a circle
when the perpendiculars drawn to
them from the centre are equal.
72
DEFINITIONS.
And the ftraight line on which the greater perpendi-
cular falls is faid to be farther from the centre.
VI.
A fegment of a circle is the figure contained
by a ftraight line and the part of the circum-
ference it cuts off.
VII.
An angle in a fegment is the angle con-
tained by two ftraight lines drawn from any
point in the circumference of the fegment
to the extremities of the ftraight line which
is the bafe of the fegment.
VIII.
An angle is faid to ftand on the part of
; the circumference, or the arch, intercepted
between the right lines that contain the angle.
IX.
A fed:or of a circle is the figure contained
by two radii and the arch between them.
DEFINITIONS.
11
Similar fegments of circles
are thofe which contain
equal angles.
Circles which have the fame centre are
called concentric circles.
74
BOOK III. PROP. I. PROB.
O Jind the centre of a given
circle
o
Draw within the circle any ftraight
Hne — ^
draw
hi left .
ma
ke.
i^MMMi ^ and the point of
biledtion is the centre.
For, if it be pofTible, let any other
point as the point of concourfe of .^— — , ---..---
and — .— — be the centre.
Becaufe in
and
■ ----— (J'^yp- ^""^ 2* I J def. 15.)
-- (conft.) and ••■- common,
^B. I, pr. 8.), and are therefore right
angles ; but
^ = ^_| (con
ft.
(ax. I I .)
which is abfurd ; therefore the aflumed point is not the
centre of the circle ; and in the fame manner it can be
proved that no other point which is not on — ^^— • is
the centre, therefore the centre is in ^— ^^— , and
therefore the point where 1 is bifedled is the
centre.
Q. E. D.
BOOK III. PROP. 11. THEOR.
75
STRAIGHT line C—)
joining two points in the
circumference of a circle
lies ivholly within the circle.
Find the centre of
o
(B.S-pr.i.);
from the centre draw
to any point in
meeting the circumference from the centre ;
draw — — — and .
Then
= -^ (B. i.pr. 5.)
but
or
CZ ^ (B. I. pr. 16.)
(B. I. pr. 19.)
but
.*. every point in
lies within the circle.
Q. E. D.
76
BOOK III. PROP. III. THEOR.
Draw
F a Jlraight line (
drawn through the centre of a
circle
o
bife£lsachord
( •'•■) which does not paj's through
the centre, it is perpendicular to it; or,
if perpendicular to it, it bifeSls it.
and
to the centre of the circle.
In >^ I and L..._V
■• •■ ■■»
and ,*,
m^^^ common, and
= (B. 1. pr. 8.)
_L -..«.- (B. I. def. 7.)
Again let
Then in
J and L^„..T^
(B. i.pr. 5.)
(hyp.)
and
and .*.
(B. I. pr. 26.)
bifedts
Q. E. D.
BOOK HI. PROP. IF. THEOR.
11
F in a circle tiaojlraight lines
cut one another, which do
not both pafs through the
centre, they do not bifeSl one
another.
If one of the lines pafs through the
centre, it is evident that it cannot be
bifecfted by the other, which does not
pafs through the centre.
But if neither of the Hnes — =— ^^— or •— ^-i—
pafs through the centre, draw ——----.
from the centre to their interfedlion.
If «i^^^ be bileded, ._._._ _L to it (6. 3. pr. 3.)
.*. ^^ = I ^ and if — be
bifed:ed,
(B. 3- P'-- 3-)
and .*,
5 a part
equal to the whole, which is abfurd :
.*. — —— — and — — — •
do not biiecfl one another.
Q. E. D.
w*
78
BOOK III. PROP. V. THEOR.
F two circles
interfeSl, they have not the
(0)
Janie centre.
Suppofe it poflible that two interfedting circles have a
common centre ; from fiich fuppofed centre draw ^.i^..
to the interfering point, and ^^—^— ••■--■■
the circumferences of the circles.
meetmg
Then
and <—
(B. i.def 15.)
- (B. I. def. 15.)
«»- J a part
equal to the whole, which is abfurd :
.', circles fuppofed to interfedl in any point cannot
have the fame centre.
Q,E. D.
BOOK III. PROP. VL THEOR.
79
F tivo circles
touch
one another internally, they
have not the fame ce?itre.
For, if it be poffible, let both circles have the fame
centre; from fuch a fuppofed centre draw ---■»
cutting both circles, and ■— — ^— to the point of contadl.
Then
and —
(B. i.def. 15.)
(B. I.def. 15.)
J a part
equal to the whole, which is abfurd ;
therefore the aiTumed point is not the centre of both cir-
cles ; and in the fame manner it can be demonftrated that
no other point is.
g E. D.
8o
BOOK HI. PROP. FII. THEOR.
nCURE 1.
FIGURE II.
F Jt'om any point within a circle
which is not the centre, lines
o
are drawn to the circumference ; the greatejl of thofe
lines is that (-i^.«i"») which pajfes through the centre,
and the leaf is the remaining part ( ^ of the
diameter.
Of the others, that ( ^— ■— — > ) which is nearer to
the line pafing through the centre, is greater than that
( mmmmm^^ ) wliich Is itiore remote.
Fig. 2. The two lines (•
and
)
which make equal angles with that pafpng through the
centre, on oppofite fdes of it, are equal to each other; and
there cannot be drawn a third line equal to them, from
the fame point to the circumference.
FIGURE I.
To the centre of the circle draw —-—— and -— «-- — j
then "— -— rr —..—.. (B. i. def. 15.)
......i^Mi ^ ^^— -|- ...«>. C — — ^-» (B.I . pr. 20.)
in like manner — — — (• may be fhewn to be greater than
.i.M__- , or any other line drawn from the fame point
to the circumference. Again, by (B. i. pr. 20.)
take — — from both ; /. — — — CI ....1^— (ax.),
and in like manner it may be fhewn that — ^— ^ is lefs
BOOK III. PROP. VII. THEOR. 8i
thiin any other line drawn from the fame point to the cir-
cumference. Again, in y*/ and
common, ^^ [^ IV , and
(B. I. pr. 24.) and
may in like manner be proved greater than any other line
drawn from the fame point to the circumference more
remote from -——■-,
FIGURE II.
If ^=^. hen = ,if„o.
take — — ^— r= ^— — draw '■"■'■, then
^c— 'I :>»^
in ^^ I and I ,^^ , — — common.
(B. i.pr. 4.)
a part equal to the whole, which is abfurd :
■"— — IS * and no other line is equal to
■^ drawn from the fame point to the circumfer-
ence ; for if it were nearer to the one pafling through the
centre it would be greater, and if it were more remote it
would be lefs.
Q. E. D.
M
82
nOOK HI. PROP. nil. THEOR.
The original text of this propolition is here divided into
three parts.
I.
^Voll ^^ f''°'" '' P°'"^ without a circle, Jlraight
iiKCs \ — — — \ ore (jrd'wn to
V»y
the cir-
cu/nference ; of thofe falling upon the concave circum-
ference the greatejl is that ( — ») ichich fafja
through the centre, and the line ( i ) ichich is
nearer the greatejl is greater than that ( )
'ichich is more remote.
Draw
and .■■-..>... to the centre.
Then. ■— . which pallcs through the centre, is
greatcll; for fince — — nz , it ^-^-^—
be added to both. -•■-■ ^ — ^— -|- ;
l^iit C — — — {^- 1- P'- -^•) •'- — — — is greater
than ;inv other line dr.iwn from the fame point to the
concave circumference.
Again in
and
BOOK rrr. prop. nir. tiikor
and — ^— ^ common, Init
0
(B. I. pr. 24.);
and in like manner
may be Ihcwn ZZ t'l-"! -^'7
other line more remote from
II.
Of thofc lines falling on the convex circumference the
leaf is that (————) which being produced would
pafs through the centre, and the line which is nearer to
the lea/l is Icf than that which is more remote.
For, fince — — — -j-
and
/«
And again, fince — — -|-
h (B. i.pr. 21.),
and — — rs .
— — -, And lb of others.
III.
Alfo the lines making equal angles with that which
pajjes through the centre arc equal, whether /ailing on
the concave or convex circumfrence ; and no third line
can he drinvn equal to the/n from the f<imc point to the
circumference.
Forif •■
make
•— ^ C "■"•"", Init making ^ =: ^ ;
----- ^ ----- ^ ;iml tl|;l\V ...... -.^
84
BOOK III. PROP. Fin. THEOR.
Then in
and
) and /
;
/
we have
common, and alfo ^ =: 41,
- = (B. I. pr. 4.);
but
which is abfurd.
of
■ ■■iisBisB IS not Z!Z
----- — 9 .*. --■■
-, nor to any part
is not r~ — -----^
Neither is
'— , they are
to each other.
And any other line drawn from the fame point to the
circumference muft He at the fame fide with one of thefe
lines, and be more or lefs remote than it from the line pall-
ing through the centre, and cannot therefore be equal to it.
Q. E. D.
BOOK in. PROP. IX. THEOR.
85
F a point be taken ivithin a
circle ( ] , from which
o-
more thwi two equal Jlraight lines
can be drawn to the circumference^ that
point tnuji be the ceiitre of the circle.
For, if it be fuppofed that the point |^
in which more than two equal ftraight
lines meet is not the centre, fome other
point — .. muft be; join thefe two points by
and produce it both ways to the circumference.
Then fince more than two equal ftraight lines are drawn
from a point which is not the centre, to the circumference,
two of them at leall; muft lie at the fame fide of the diameter
.; and fince from a point
/\
w
hich
is
not the centre, ftraight lines are drawn to the circumference ;
the greateft is -i^— .--= = ^ which pafies through the centre :
and — ^— — — which is nearer to ^^■—••«'. |^ ———~-
which is more remote (B. 3. pr. 8.) ;
but — — — ^— rr ^— ^— =- (hyp.) which is abfurd.
The fame may be demonftrated of any other point, dif-
ferent from / |\^ which muft be the centre of the circle,
Q. E. D.
86
BOOK III. PROP. X. THEOR.
NE circle I } cannot interfeSl another
I J /« more points than two.
For, if it be poffible, let it interfedl in three points ;
from the centre of ( I draw
O
to the points of interfedlion ;
(B. I. def. 15.),
but as the circles interfedl, they have not the fame
centre (B. 3. pr. 5.) :
.*, the alTumed point is not the centre of
o.
and
and
are drawn
from a point not the centre, they are not equal (B. 3.
prs. 7, 8) ; but it was fhewn before that they were equal,
which is abfurd ; the circles therefore do not interfedl in
three points.
Q. E. D.
BOOK III. PROP. XL THEOR.
87
F two circles
o
o
and
touch one another
internally, the right line joining their
centres, being produced, Jliall pafs through
a point of contaSl.
For, if it be poffible, let
join their centres, and produce it both
ways ; from a point of contadl draw
— — — to the centre of ( J , and from the fame point
of contadl draw -•-•--— to the centre of I J .
Becaufe in
4
+
■■■•■•■•t
(B. I . pr. 20.),
and
o
as they are radii of
8B BOOK III. PROP. XI. THEOR.
but — ^ -|- I rr — — — ; take
away — — — which is common,
hut — i^— i ^ -- — --^
ii of r^ ,
becaufe they are radi
and ,*, --»-" C ^^ ^ P^i't greater than the
whole, which is abfurd.
Tlie centres are not therefore fo placed, that a line
joining them can pafs through any point but a point of
contadt.
Q. E. D.
BOOK III. PROP. XII. THEOR.
89
F two circles
o
t/ier externally, the Jiraight line
1 1 joining their centres,
pajfes through the point of contaB.
touch one a7io
If it be polTible, let
join the centres, and
not pafs through a point of contadl ; then from a point of
contad: draw -"^^^== and ""■••^■^-' '-• to the centres.
Becaufe
and .
and •
. +
(B. I. pr. 20.),
= (B. I. def. 15.),
= ^ (B. I. def. 15.),
+
', a part greater
than the whole, which is abfurd.
The centres are not therefore fo placed, that <-he line
joining them can pafs through any point but the point of
contadl.
Q.E. D.
90 BOOK in. PROP. XIIL THEOR.
FIGURE I.
FIGURE II.
FIGURE III.
NE circ/e can-
not touch ano-
ther, either
externally or
internally, in more points
than one.
Fig. I . For, if it be poffible, let
and f 1 touch one
o
another internally in two points ;
draw — — - joining their cen-
tres, and produce it until it pafs
through one of the points of contadl (B. 3. pr. 11.);
draw — — — and •^— ^—^ ,
.-. if
(B. I. def 15.),
be added to both,
+
but
and .*.
+
+
which is abfurd.
.■ (B. I. def 15.),
- = — _— ; but
— — (B. I. pr. 20.),
BOOK III. PROP. XIII. THEOR. 91
Fig. 2. But if the points of contadl be the extremities
of the right line joining the centres, this ftraight line muft
be bifedled in two different points for the two centres; be-
caufe it is the diameter of both circles, which is abfurd.
, let f j and I J
Fig. 3. Next, if it be poffible
touch externally in two points; draw — — joining
the centres of the circles, and paffing through one of the
points of contact, and draw — i— — — and -^^— — ,
— -^ z= _ (B. I. def. 15.);
nd ...«■■•. zr I (B. i. def. 15.);
-\- — — -^ Z:Z. ■BMBsaaa * but
+ ^-^™''— C ».-- (B. I. pr. 20.),
which is abfurd.
There is therefore no cafe in which two circles can
touch one another in two points.
Q E. D.
92
BOOK III. PROP. XIV. THEOR.
Then
and
infcribed in a circle are e-
qually dijiantfrom the centre ;
andalfofjlraight lines equally
dijiafit from the centre are equal.
From the centre o
o
draw
-L
to —
,join
■••■■ and --•
^— and — ■
fince
= half '" (B. 3. pr. 3.)
= i (B. 3-pr-3-)
= — (hyp.)
and
(B. I. def. 15.)
and
but fince >- s^ is a right angle
= ' + MB.i.pr.47-)
.' = ' + ^ for the
.% -^ +
fame reafon.
BOOK III. PROP. XIV. THEOR. 93
Alfo, if the lines 1 ..■.•»■ and •— i»«...«.r be
equally diflant from the centre ; that is to fay, if the per-
pendiculars -■-■•■■■■■■ and -m........ be given equal, then
For, as in the preceding cafe,
. ^ :::= __i.^, and the doubles of thefe
....... and ^n....... are alfo equal.
Q. E. D.
94
BOOK III. PROP. XV. THEOR.
FIGURE I.
but
HE diameter is the greatejl Jiraight
line in a circle : and, of all others,
that which is neareji to the centre is
greater than the more remote.
FIGURE I.
The diameter ^^ is CZ any line
For draw ' and ••••••••••
Then .••■>••»■■> ^ ^— ^._i
and •^— ^— = — — — .
mXm ■■■■•»««» ^^
■■■•■■•■•
(B. I . pr. 20.)
Again, the Hne which is nearer the centre is greater
than the one more remote.
Firft, let the given lines be — — ^ and - ,
wnich are at the fame fide of the centre and do
not interfed: ;
draw J '
BOOK III. PROP. XF. THEOR.
95
FIGURE II.
Let the given lines be ■^— ^ and ^i— ^
which either are at different fides of the centre,
orinterfedt; from the centre draw - -■•--
and -»-«-—- J_ -^^Mi->i» and ,
FIGURE II.
make
draw
Since
and
the centre,
but — — —
and
are equally diflant from
(B. 3. pr. 14.);
(Pt. i.B. 3.pr. 15.),
Q. E. D.
96
500 A' ///. PROP. XVI. THEOR.
llEJiraiglit
line -
draii-n
from the
extremity of the diame-
ter
of a
circle
h
perpendicular to it falls
••.^^ ^,., without the circle.
||» • '^ * And if anyjlraight
*** line ........ be
drawn from a point
————— within that perpendi-
cular to the point of contaB, it cuts the circle.
PART I
If it be poffible, let ^ which meets the circle
again, be _L , and draw ,
Then, becaufe
^ = ^ (B. i.pr. 5.),
and .'. each of these angles is acute. (B. i. pr. 17.)
but ^^ =r I J (hyp.), which is abfurd, therefore
ii...._ drawn _L — ^^^— does not meet
the circle again.
BOOK in. PROP. XVI. THEOR. 97
PART 11.
Let — Bi*—"— be _L -^-^^— and let -— — - be
drawn from a point y between — ■— ■— • and the
circle, which, if it be poflible, does not cut the circle.
Becaufe H^ =: | ^ ,
.*. ^^ is an acute angle ; fuppofe
....... ...4.... _L .-■«•-•-, drawn from the centre of the
circle, it mufl; fall at the fide of ^^ the acute angle.
,*, B^^ which is fuppofed to be a right angle, is C ^^;
but ............ = ,
and .*, --••••.. ^ ......«..■—, a part greater than
the whole, which is abfurd. Therefore the point does
not fail outfide the circle, and therefore the ftraight line
■ ••••MiM* cuts the circle.
Q. E. D.
98
BOOK III. PROP. XVII. THEOR.
O Jraiv a tangent to a given
circle \ \ from a
o
given point, either in or outjide of its
'•♦^ circumference.
If the given point be in the cir-
cumference, as at I , it is plain that
the ftraight line "■■" J_ -«— — -
the radius, will be the required tan-
gent (B. 3. pr. 16.) But if the given point ^
outlide of the circumference, draw —
from it to the centre, cutting
be
draw •■««■■■■** ^_
concentric with
then
o
.., defcribe
radius^ .■■■■ub^,
will be the tangent required.
BOOK III. PROP. XFII. THEOR.
99
XV
/
/
For
in
and i\.
, ^^^ common,
(B. I. pr. 4.) flB =: ^^^ ^ a right angle,
.*. ^a^a^B is a tangent to
Q. E. D.
o
loo BOOK III. PROP. XFIII. THEOR.
and .*,
F a right line •• — be
a tangent to a circle, the
fir aight line — ^— draivn
from the centre to the
i point of contaSl, is perpendicular to it.
For, if it be poflible,
let *>■ be ^ — ...
then becaufe
= [^
is acute (B. i . pr. 17.)
c
(B. I. pr. 19.);
but
»•*•■ , a part greater than
the whole, which is ablurd.
/, .»«.. is not _L -"•—•••5 and in the fame man-
ner it can be demonftrated, that no other line except
— ■— — is perpendicular to «•-.■...-• ,
Q. E. D.
BOOK III PROP. XIX. THEOR.
lOI
F a Jlra'tght line
be a tangent to a circle,
thejiraight line ,
drawn perpendicular to it
from point of the contact, pajfes through
the centre of the circle.
For, if it be pofTible, let the centre
be without «
and draw
• ••- from the fuppofed centre
to the point of contadl.
Becaufe
X
(B. 3.pr. i8.)
.'. ^^ =: I Ji , a right angle ;
but ffj^ = I 1 (hyp.)' and /. ^ =
a part equal to the whole, which is abfurd.
Therefore the affumed point is not the centre ; and in
the fame manner it can be demonftrated, that no other
point without _„_^ is the centre.
Q. E. D.
102
BOOK III. PROP. XX. rUEOR.
FIGURE I
HE angle at the centre of a circle, is double
the angle at the circumference, when they
have the fame part of the circumference for
their bafe.
FIGURE I.
Let the centre of the circle be on
a fide of ^ ,
Becaufe
i = ^
= ^ (B. i.pr.5.).
But
+
\
or
=: twice . (B. i. pr. 32).
FIGURE II.
FIGURE II.
Let the centre be within ^ ^ the angle at the
circumference ; draw ■■■^^^— from the angular
point through the centre of the circle ;
then ^ := r ? and = ^^ ,
becaufe of the equality of the fides (B. i. pr. 5).
BOOK III. PROP. XX. THEOR. 103
Hence
-|- ^ -|~ "I" ^ twice
But ^ = ^ + ^ , and
r= twice
FIGURE III.
Let the centre be without W and
FIGURE III.
draw m^
the diameter.
B(
jcaufe ▼
: twice ^ ; i
r=
twice
^^ (cafe I .) ;
•
• •
A
^ twice ▼ ,
Q. E. D.
I04 BOOK III. PROP. XXI. THEOR.
FIGURE I.
HE angles ( ^^ , ^^ ) in the fame
Jegment of a circle are equal.
FIGURE I.
Let the fegment be greater than a femicircle, and
draw — ^^^— and — — — — to the centre.
twice ^^ or twice
(B. 3. pr. 20.),-
4 = 4
4
FIGURE II.
FIGURE II.
Let the fegment be a femicircle, 01 lefs than a
femicircle, draw —— — the diameter, alfo draw
^=4a„dV = ^
(cafe I.)
Q. E. D.
J
BOOK III. PROP. XXII. THEOR. 105
HE oppojite arigJes
Af
and ^^ . ^^1 and
^r of any quadrilateral figure in-
fcr'ibed in a circle, are together equal to
two right angles.
Draw
and
the diagonals ; and becaufe angles in
the fame fegment are equal ^W =: ^^
and ^r rr: ^^ |
add ^ to both.
two right angles (B. i. pr. 32.). In like manner it may
be Ihown that,
Q. E. D.
io6 BOOK III. PROP. XXIII. THEOR.
PON t/ie fame
Jlraight line,
and upon the
fame fide of it,
two fmilar fegments of cir-
cles cannot be conflruBed
which do not coincide.
For if it be poflible, let two fimilar fegments
o
and
be conftrudled ;
draw any right line
draw «
• cutting both the fegments,
and ^^-HMM .
Becaufe the fegments are fimilar.
(B. 3. def 10.),
but ^M [Z ^^ (B. I. pr. 16.)
which is abfurd : therefore no point in either of
the fegments falls without the other, and
therefore the fegments coincide.
Q. E. D.
BOOK III. PROP. XXIV. THEOR.
107
IMILAR
fegments
and
, of cir-
cles upon equal Jlraight
lines ( '^^^ and — — )
are each equal to the other.
For, if
that —
be fo applied to
- may fall on ^— ^—
may be on the extremities
the extremities of
and
at the fame fide as
becaufe
muft wholly coincide with
and the fimilar fegments being then upon the fame
flraight line and at the fame fide of it, muft
alfo coincide (B. 3. pr. 23.), and
are therefore equal.
Q. E. D.
io8
BOOK III. PROP. XXV. PROP.
SEGMENT of a circle
being given, to defcribe the
circle of 'which it is the
feginent.
From any point in the fegment
draw ^^— ^ and — ^^^^ bifeft
them, and from the points of biledlion
draw
and
where they meet is the centre of the circle.
Becaufe — ..__ terminated in the circle is bifedied
perpendicularly by ^■"■■"^ , it palTes through the
centre (B. 3. pr. i.), likewife ^a^^M^ pafles through
the centre, therefore the centre is in the interfedlion of
thefe perpendiculars.
Q.E. D.
BOOK III. PROP. XXVI. THEOR. 109
N egua/ circles
the arcs
O ""' o
on which
Jiand equal angles, whether at the
centre or
circum
ference, are equal.
Firfl, let ^^
at the
centre.
Then fince
0 =
mmm ^
O-
/\
and ^♦;;.„
■•\
have
■ ■■■■■■
and
But
▲ =▲
(B. i.pr.4.).
(B.3.pr. 20.);
• • O '"' o
are fimilar (B. 3. def. 10.) ;
they are alfo equal (B. 3. pr. 24.)
110 BOOK III. PROP. XXVI. THEOR.
If therefore the equal fegments be taken from the
equal circles, the remaining fegments will be equal ;
hence
(ax. 3.);
and .*,
But if the given equal angles be at the circumference,
it is evident that the angles at the centre, being double
of thofe at the circumference, are alfo equal, and there-
fore the arcs on which they fland are equal.
Q. E. D.
BOOK III. PROP. XXVn. THEOR. 1 1 1
N equal circles.
O-O
the angles and ^^ which Jiand upon equal
arches are equal, whether they be at the centres or at
the circumferences.
For if it be poffible, let one of them
▲
be greater than the other
and make
▲
\ = 4
/. V_^-" = **»n„..« (B. 3. pr. 26.)
but V«^ = **♦.....•♦ (hyp.)
.'. ^-i_ ^ := >fc^ _^^^ a part equal
to the whole, which is abfurd ; .*, neither angle
is greater than the other, and
,*, they are equal.
Q. E. D.
••••.•■•••
112 BOOK III. PROP. XXVIII. THEOR.
N equa/ circles
o-o
egual chords
arches.
cut off equal
From the centres of the equal circles.
draw
and
and becaufe
c=o
alfo
(IW-)
(B. 3. pr. 26.)
and
.0 = 0
(ax. 3.)
Q. E. D.
BOOK III. PROP. XXIX. THEOR. 113
N equal circles
nd ••-- ivhich fub~ \ ^ ^^ /
the chords ^— -^^ and
tend equal arcs are equal.
If the equal arcs be femicircles the propofition is
evident. But if not,
let — ^^i^ . — — i^ , and
be drawn to the centres ;
becaufe
and
but ^— — ^ and
(hyp.)
(B.3.pr.27.);
•■»...... and «•-'
(B. I. pr. 4.);
but thefe are the chords fubtending
the equal arcs.
Q. E. D.
114
BOOK III. PROP. XXX. PROB.
O l>ife^ a given
n-
arc
draw
Draw
make — ^
_L — ^^-■" , and it bifedls the arc.
■*•«•■•■
Draw •"••■"»■ and
and
— --— (confl:.),
is common,
(conft.)
. (B. i.pr.4.)
= y"*'\ (B. 3- pr. 28.).
and therefore the given arc is bifeded.
Q. E. D.
BOOK III. PROP. XXXI. THEOR. 115
N a circle the angle in afemicircle is a right
angle, the angle in a fegment greater than a
femicircle is acute, and the angle in a feg-
ment lefs than a femicircle is obtufe.
FIGURE I.
FIGURE I.
The angle ^ in a femicircle is a right angle.
V
Draw
and
and
V
= ^ (B. i.pr. 5.)
+ A= V
^ the half of two
right angles sz a right angle. (B. i. pr. 32.)
FIGURE II.
The angle ^^ in a fegment greater than a femi-
circle is acute.
FIGURE II.
Draw
the diameter, and .-
^ a right angle
^^ is acute.
ii6 BOOK III. PROP. XXXI. THEOR.
FIGURE III.
FIGURE III.
The angle ^^^^ in a fegment lefs than femi-
circle is obtufe.
Take in the oppofite circumference any point, to
which draw «mmm* and .
^
Becaufe W^ -|-
(B. 3. pr. 22.)
^Oh
but
(part 2.),
is obtufe.
Q. E. D.
BOOK III. PROP. XXXIL THEOR. 117
F a rig/it line ^■■■■ii"— ■
be a tangent to a circle,
and frotn the point of con-
tact a right line "
be drawn cutting the circle, the angle
jg^ made by this line with the tangent
is equal to the angle ^^ in the alter-
ate fegment of the circle.
If the chord fhould pafs through the centre, it is evi-
dent the angles are equal, for each of them is a right angle.
(B. 3. prs. 16, 31.)
But if not, dra'V
from the
point of contadl, it muft pafs through the centre of the
circle, (B. 3. pr. 19.)
.-. ^ = ^ (B.3.pr.3i.)
W + f =• CA = f (B- I- pr. 32.)
/. ^ = ^ (ax.).
Again CJ = iV\ = _ + ^
(B. 3. pr. 22.) ^
/. C. y = ^m » (ax.), which is the angle in
the alternate fegment.
Q. E. D.
ii8 BOOK III. PROP. XXXIII. PROP.
N a given ftraight line ^^^—
to dejcribe a Jegment of a
circle that Jliall contain an
angle equal to a given angle
^,ty,
If the given angle be a right angle,
bifedt the given line, and defcribe a
femicircle on it, this will evidently
contain a right angle. (B. 3. pr. 31.)
If the given angle be acute or ob-
tufe, make with the given line, at its extremity.
, draw
make
with
f
defcribe
and
or — ' ■ ■ ■ as radius,
for they are equal.
is a tangent to
o
(B. 3. pr. 16.)
divides the circle into two fegments
capable of containing angles equal to
/ W and j/^ which were made refpedlively equal
and
(B. 3.pr. 32.)
Q. E. D.
BOOK III. PROP. XXXIV. PROP. 119
O cut off from a given cir-
cle
o
a fegment
which fiall contain an angle equal to a
given angle
I>raw
(B. 3. pr. 17.),
a tangent to the circle at any point ;
at the point of contad: make
and
>
the given angle ;
contains an angle ^ the given angle.
Becaufe
and «
angle in
>
• IS a tangent,
cuts it, the
(B. 3. pr. 32.),
but
(conft.)
Q. E. D.
120 BOOK III. PROP. XXXV. THEOR.
FIGURE I.
FIGURE II.
F two chords \ ••••"" i .^ ^ circle
interfeSl each other, the reBangle contained
by the fegments of the one is equal to the
re 51 angle contained by the fegments of the other.
FIGURE I.
If the given right lines pafs through the centre, they are
bifedled in the point of interfed:ion, hence the recftangles
under their fegments are the fquares of their halves, and
are therefore equal.
FIGURE II.
Let — "— - pafs through the 'centre, and
.«■>■.■- not; draw — — — — and .
Then
X
or
» (B. 2. pr. 6.),
X = '
X - =
■■ (B. 2. pr. 5.).
X
FIGURE III.
FIGURE III.
Let neither of the given lines pafs through the
centre, draw through their interfedlion a diameter
........ 9
and X = X
>■■■•■ (Part. 2.),
alfo - X = X
........ (Part. 2.) ;
Q. E. D.
BOOK III. PROP. XXXFI. THEOR.
121
F from a point without a FIGURE I.
circle twojlraight lines be
drawn to it, one of which
— ■'^"» is a tangent to
the circle, and the other — — --
cuts it ; the re^angle under the whole
cutting line — ••«■• and the
external fegment ^-^ is equal to
the fquare of the tangent -^— ,
FIGURE I.
Let — i— •• pafs through the centre;
draw from the centre to the point of contadl ;
- (B. i.pr. 47),
minus
or
mmus
(B.2.pr. 6).
FIGURE II.
If ■"•'■ do not
pafs through the centre, draw
FIGURE n.
and
Tl
len
"X
minus
(B. 2. pr. 6), that is.
mmus
.2
(B. 3.pr. 18).
Q. E. D.
122 BOOK in. PROP. XXXVII. THEOR.
but
F from a point outfide of a
circle tivoftraight lines be
draivn, the one -■^-■»
cutting the circle, the
other — — ^ meeting it, and if
the reSiangle contained by the whole
cutting line —"« and its ex-
ternal fegment ■-..—.. be equal to
thefquare of the line meeting the circle,
the latter .m.^m^m,—> is a tangent to
the circle.
Draw from the given point
^— , a tangent to the circle, and draw from the
centre .«»■», ••».•«.••, and — --- — -^
-■^ = X (B.3.pr.36.)
2 = X (i^yp-).
and
Then in ',
and
and
and -,^^^
...a»— and
is common.
but
and .*.
^ = ^ (B. i.pr. 8.);
^ ^^^ a right angle (B. 3. pr. 18.),
^r := ^_J| a right angle,
■^ is a tangent to the circle (B. 3. pr. 16.'
Q. E. D.
BOOK IV.
DEFINITIONS.
I.
RECTILINEAR figure is
faid to be infcribed in another,
when all the angular points
of the infcribed figure are on
the fides of the figure in which it is faid
to be infcribed.
II.
A FIGURE is faid to be defcribed about another figure, when
all the fides of the circumfcribed figure pafs through the
angular points of the other figure.
III.
A RECTILINEAR figure is faid to be
infcribed in a circle, when the vertex
of each angle of the figure is in the
circumference of the circle.
IV.
A RECTILINEAR figure is faid to be cir-
cumfcribed about a circle, when each of
its fides is a tangent to the circle.
124 BOOK IF. DEFINITIONS.
V.
A CIRCLE is faid to be tnfcribed in
a redlilinear figure, when each fide
of the figure is a tangent to the
circle.
VI.
A CIRCLE is faid to be circum-
fcribed about a redtihnear figure,
when the circumference pafles
through the vertex of each
angle of the figure.
y
is circumfcribed.
VII.
A STRAIGHT line is faid to be tnfcribed in
a circle, when its extremities are in the \
circumference.
The Fourth Book of the Elements is devoted to the folution of J
problems t chiefly relating to the infcription and circumfcrip-
tion of regular polygons and circles.
A regular polygon is one whofe angles and fides are equal.
BOOK IF. PROP. I. PROP.
125
N a given circle
O
to place ajlraight line,
equal to agivenjlraight line ( ),
not greater than the diameter of the
circle.
Draw
, the diameter of
and if ..-....^— . ^:z
', then
the problem is folved.
But if
be not equal to
(hyp-) ;
make
(B. I. pr. 3.) with
as radius.
defcribe I ), cutting f |, and
draw ^ which is the line required.
For
(B. I. def. 15. confl.)
Q. E. D.
126
BOOK IF. PROP. II. PROP.
N a given circle
O
to m-
Jcribe a triangle equiangular
to a given triangle.
To any point of the given circle draw
, a tangent
(B. 3. pr. 17.) ; and at the point of contadt
make ^^^ — - ^^ (B. i. pr. 23.)
and in like manner
draw
Ik
and
Becaufe
and
J^ = ^ (conft.)
Jg^ = ^^ (B. 3. pr. 32.)
.*. ^^ = ^U ; alfo
V^ =: ^r ^°^ ^^ i-3xtit reafon.
,\^ = ^ (B. i.pr. 32.),
and therefore the triangle infcribed in the circle is equi-
angular to the given one.
Q^E. D.
BOOK IF. PROP. III. PROB.
127
BOUT a given
circle
O
to
circumfcribe a triangle equi-
angular to a given triangle.
Produce any fide
, of the given triangle both
ways ; from the centre of the given circle draw
any radius.
Make ^ft =
^
and
(B. I. pr. 23.)
r=%
At the extremities of the three radii, draw
and .-.-...--, tangents to the
given circle. (B. 3. pr. 17.)
Zi
The four angles of >^Wi ^B , taken together, are
equal to four right angles. (B. i. pr. 32.)
128 BOOK IF. PROP. III. PROB.
but ^B ^"d ^^^ ^^^ I'ight angles (confl.)
two right angles
but ^H^ ^ La^^^M^ (B' ^' P''- ^3-)
and = ^^ (conft.)
and ,*,
In the fame manner it can be demonftrated that
<^=^,
4 = 4
(B. i.pr. 32.)
and therefore the triangle circumfcribed about the given
circle is equiangular to the given triangle.
Q, E. D.
i
BOOK IF. PROP. IV. PROB.
129
fcribe a circle.
Bifeft
^ and ^V.
(B. i.pr. 9.) by
and "— ^^
from the point where thefe lines
meet draw -•■■— ,
and »•••■ refpedlively per-
pendicular to — BMI^HiB ,
In
and
/
and
>
A 4
and
common, ,*, «••••••.•. ^^ .■■■...•.». (B. i. pr. 4and 26.)
In like manner, it may be fhown alfo
that
hence with any one of thefe lines as radius, defcribe
and it will pafs through the extremities of the
o
other two ; and the fides of the given triangle, being per-
pendicular to the three radii at their extremities, touch the
circle (B. 3. pr. 16.), which is therefore infcribed in the
given circle.
Q. E. D.
130
BOOK IV. PROP. V. PROB.
O defer ibe a circle about a given triangle.
■" and
--- (B. I. pr. 10.)
From the points of bifedlion draw —
■•■■•■■••• J_ -^■~— ^ and '
and
refpec-
tively (B. i. pr. 11.), and from their point of
concourfe draw i^--^^^, «••■-—— and
and defcribe a circle with any one of them, and
it will be the circle required.
In
(conft.).
- common,
^ (conft.),
(B. I. pr. 4.).
■■■^•■■aiaKa
In like manner it may be fhown that
, , ■■■■««■■■■ ^^ ^■^■^^^^^^" m^^^ "^^^^ \ and
therefore a circle defcribed from the concourfe of
thefe three lines with any one of them as a radius
will circumfcribe the given triangle.
Q. E. D.
BOOK IF. PROP. FI. PROB. 131
O
N a given circle f j /<?
infcribe a fquare.
Draw the two diameters of the
circle _L to each other, and draw
o
is a fquare.
f
For, iince ^^^^ and ^^^ are, each of them, in
a femicirclc, they are right angles (B. 3. pr. 31),
/. — ^ 11 (B. i.pr. 28):
and in like manner
And becaufe mg ^ |^^ (confl.), and
•••■•■■SM« """ >■■>■■■■■■■ ""• •••»»•■•■••= f B. I . def. I c).
.*. — = —> — (B. I. pr. 4);
and fmce the adjacent fides and angles of the parallelo-
gram ^ X are equal, they are all equal (B. i. pr. 34) ;
o
and /, -^ ^ , infcribed in the given circle, is a
fquare. Q. E. D.
132
BOOK IF. PROP. VIL PROP.
BOUT a given circle
I i ^^ circumfcribe
a fquare.
Draw two diameters of the given
circle perpendicular to each other,
and through their extremities draw
1 9 9
tangents to the circle ;
and —
and
D
is a fquare.
— / I a right angle, (B. 3. pr. 18.)
alfo
- II
(conft.),
5 in the fame manner it can
be demonftrated that
that — ^^ and -
• •■*••« ■■ !■
and alfo
,», I I is a parallelogram, and
becaufe
they are all right angles (B. i. pr. 34)
it is alfo evident that — --^ , —— ^.^ ,
and -i^— ^ are equal.
D
is a fquare.
Q. E. D.
BOOK IV. PROP. Fill. PROB.
133
O infcribe a circle in a
given fquare.
Make
and
draw
and — — II ..
(B. I. pr. 31.)
and fince
is a parallelogram ;
is equilateral (B. i. pr. 34.)
In like manner, it can be fhown that
are equilateral parallelograms ;
and therefore if a circle be defcribed from the concourfe
oi thefe lines with any one of them as radius, it will be
infcribed in the given fquare. (B. 3. pr. 16.)
CLE. D.
134
BOOK IF. PROP. IX. PROB.
O defcribe a circle about a
given fquare
3
Draw the diagonals ^— — .-.
and "— ■ interfedting each
other ; then,
becaufe
"^-^Ik
have
their fides equal, and the bafe
■ ■»«*»■• comnion to both,
or
r
^
(B. i.pr. 8),
is bifedled : in like manner it can be (hown
that
is bifedted ;
hence
^k rr ^^ their halves.
•. = ; (B. I. pr. 6.)
and in like manner it can be proved that
If from the confluence of thefe lines with any one of
them as radius, a circle be defcribed, it will circumfcrihe
the given fquare.
Q. E. D.
BOOK IV. PROP. X. PROP.
'35
O conftruSl an ifofceles
triangle, in which each of
the angles at the bafejliail
he double of the vertical
angle.
Take any fliaiwht line ^—
and divide it fo that
X =
(B. 2. pr. I I.)
With —I"" as radius, defcribe
o
in it from the extremity of the radius,
(B. 4. pr. i) ; draw
Then
\
and place
\ is the required triangle.
For, draw
and defcribe
O
Since
about ^ I (B. 4. pr. ^.)
X
• ■■■■■ X "■
•— is a tangent to ( ) (B. 3. pr. 37.)
.% m = ^ (B. 3. pr. 32),
1 36 BOOK IV. PROP. X. PROB.
add ^F to each,
••• A + < = ^ + ^;
but ▼ + A or # =z A (B. I. pr. 5) :
fince -—"m-m ^ ■"»» (B. I. pr. 5.)
confequently jH[^ ^ ^Xi ^ ^f ^ JH^
(B. I. pr. 32.)
.'. — = (B. i.pr. 6.)
.•. — ^— =z — — ^— ^ .^_. (conft.)
.-. -^ = W (B. I.pr. 5.)
.-. A=^ = A = ^ +
=: twice y^t 5 and confequently each angle at
the bafe is double of the vertical angle.
Q. E. D.
BOOK IV. PROP. XL PROB.
137
N a given circ/e
o
to infcribe an equilateral and equi-
angular pentagon.
Conftrudl an ifofceles triangle, in
which each of the angles at the bafe
fhall be double of the angle at the
vertex, and infcribe in the given
▲
circle a triangle ^^ equiangular to it ; (B. 4. pr. 2.)
^ and ^\ (B. i.pr.9.)
Bifedl
draw
and
Becaufe each of the angles
A.^.A
^
and \\ are equal,
the arcs upon which they ftand are equal, (B. 3. pr. 26.)
and .*.
and
....■■». which fubtend thefe arcs are equal (B.3.pr. 29.)
and ,*, the pentagon is equilateral, it is alfo equiangular,
as each of its angles ftand upon equal arcs. (B. 3. pr. 27).
Q^E. D.
138
BOOK IV. PROP. XII. PROP.
O defcribe an equilateral
and equiangular penta-
gon about a given circle
O-
Draw five tangents through the
vertices of the angles of any regular
pentagon infcribed in the given
circle
o
(B. 3. pr. 17).
Thefe five tangents will form the required pentagon.
Draw
' 1:™
In
and
■ ^■•■■■■■B
(B. i.pr.47),
and ■ common ;
.-.7 =
<-
twice
and ▼ = .4. (B. i.pr. 8.)
, and ^^1 =r twice ^
In the fame manner it can be demonilrated that
^^/ =: twice ^^ , and ^r ^ t^vice ^;
but ^ ^ " B. 3.pr. 27),
BOOK IV. PROP. XII. PROB. 139
their halves = j^ , alfo £ I ^ I \ 9 and
-■■■-■■■» common ;
and «-i-iaMaiii. ^ ...HMMiB,
,•, •■^-■» .— ^K ^ twice — ^— 5
In the fame manner it can be demonftrated
that 1^^^---— ^ twice ■-■^-•,
but — — = — — •
In the fame manner it can be demonftrated that the
other fides are equal, and therefore the pentagon is equi-
lateral, it is alfo equiangular, for
^^ ^ twice 1^^ and \^^ = twice j^^ ,
and therefore
• mKkl — uflB 9 1" the fame manner it can be
demon ftrated that the other angles of the defcribed
pentagon are equal.
QE. D
140
BOOK IF. PROP. XIII. PROP.
Draw
Becaufe
and
O infcribe a circle in a
given equiangular and
equilateral pentagon.
^^^ «■/ ^^ ^ given equiangular
and equilateral pentagon ; it is re-
quired to infcribe a circle in it.
Make
^=^,andi|^=^
(B. i.pr. 9.)
= - ,r=A,
common to the two triangles
&c.
/
and >A ...,.lk ;
.. and ^r ^ J|^ (B. i. pr. 4.)
And becaufe ^^ ^
,*, r= twice
4
twice
is bifedled by
In like manner it may be demonftrated that
^
IS
«••• "j and that the remaining angle of
bifedled by
the polygon is bifedled in a fimilar manner
BOOK IV. PROP. XIII. PROB. 141
Draw ^i— -i^ , -....-.. , 6cc. perpendicular to the
fides of the pentagon.
Then in the two triangles ^^ and
A
we have ^^ z= ^^^,(conft.), ^^^^i^ common,
and ^V :^ JIh = a right angle ;
, (B. I. pr. 26.)
In the fame way it may be fhown that the five perpen-
diculars on the fides of the pentagon are equal to one
another.
o
Defcribe X^ ^ with any one of the perpendicu-
lars as radius, and it will be the infcribed circle required.
For if it does not touch the fides of the pentagon, but cut
them, then a line drawn from the extremity at right angles
to the diameter of a circle will fall within the circle, which
has been fhown to be abfurd. (B. 3. pr. 16.)
f^E. D.
14*
BOOK IV, PROP. Xn\ PROB.
pO dcfirihc j r.-TiV chcn: s
grom egh:.s:
'sJ ^nd csiii
oik and ^^
Bilect ^JHk and
bT ••••••»»»«» and ..-•...... , and
^om the point of fedion, draw
._^B , >••»»• , and ^^^^ .
(B. i.pr.6):
I" like manner it mar be proved that
■ ^ ^iB^^M ^ ^^— — , and
therefore -••••••• ^ — ^— ^ ••m>»...w.
Therefore if a circle be defcribed from the point where
thefe five lines meet, with any one of tfaem
as a radius, it will circumicribe
the given pcntagoo.
Q E- D.
BOOK W. PROP. XV. PROB.
O infcribe an equilateral and equian-
gular hexagon in a gircen circle
H3
O-
From any point in the circumference of
the given circle defcribe ^ J palling
o
through its centre, and draw the diameters
and
draw
...._»■_ J .-..-..-^ .........J &c. and the
required hexagon is inicribed in the given
circle.
Since
of the circles.
palles through the centres
and
are equilateral
4 = ^
triangles, hence ^^ ^ ^^ ^ one-third of two right
angles; i^B. i. pr. 32) but
(B. i.pr. 13);
= m
^ one-third of
£Di
(B. I. pr. 32% and the angles vertically oppoiite re :::ei"e
are all equal to one another (B. i. pr. i ;\ and iland on
equal arches (B. 3. pr. 26), which are fubtended by equal
chords (B. 3. pr. 29) ; and fince each of the angles of the
hexagon is double of the angle of an equilateral triangle,
it is alio equiangular. O ^ F)
'44
BOOK IV. PROP. XVI. PROP.
O infcribe an equilateral and
equiangular quindecagon in
a given circle.
and
be
the fides of an equilateral pentagon
infcribed in the given circle, and
««»-— the fide of an inscribed equi-
lateral triangle.
The arc fubtended by
. and __
_6_
I 4
of the whole
circumference.
The arc fubtended by 1
_5_
1 i
Their difference =: tV
,'. the arc fubtended by
the whole circumference.
of the whole
circumference.
zz. tV difference of
Hence if firaight lines equal to ..-«.—■« be placed in the
circle (B. 4. pr. i), an equilateral and equiangular quin-
decagon will be thus infcribed in the circle.
Q. E. D.
BOOK V.
DEFINITIONS.
I.
LESS magnitude is faid to be an aliquot part or
fubmultiple of a greater magnitude, when the
lefs meafures the greater ; that is, when the
'^ lefs is contained a certain number of times ex-
adlly in the greater.
II.
A GREATER magnitude is faid to be a multiple of a lefs,
when the greater is meafured by the lefs ; that is, when
the greater contains the lefs a certain number of times
exadlly.
III.
Ratio is the relation which one quantity bears to another
of the fame kind, with refpedl to magnitude.
IV.
Magnitudes are faid to have a ratio to one another, when
they are of the fame kind ; and the one which is not the
greater can be multiplied fo as to exceed the other.
TAe of her definitions will be given throughout the book
where their aid is Jirjl required.
u
146
AXIOMS.
QUIMULTIPLES or equifubmultiples of the
fame, or of equal magnitudes, are equal.
If A = B, then
twice A ^ twice B, that is,
2 A = 2 B;
3Az=3B;
4 A = 4B;
&c. &c.
and i of A = i of B ;
i of A = i of B ;
&c. &c.
II.
A MULTIPLE of a greater magnitude is greater than the fame
multiple of a lefs.
Let A C B, then
2 AC 2 B;
3 ACZ3B;
4 AIZ4B;
&c. &c.
III.
That magnitude, of which a multiple is greater than the
fame multiple of another, is greater than the other.
Let 2 A C 2 B, then
ACB;
or, let 3 A C 3 B, then
ACB;
or, let w A CZ m B, then
ACB.
\
BOOK V. PROP. I. THEOR.
H7
F any number of magnitudes be equimultiples of as
many others, each of each : what multiple soever
any one of the firjl is of its part, the fame multiple
jhall of the fir ft magnitudes taken together be of all
the others taken together.
LetQQQQQ be the fame multiple of Q,
that Pip^^^ isof ^.
that OOOQO ^s of Q.
Then is evident that
• QQQQQ
OOOOQ
fQ
is the fame multiple of <
Q
which that QQQQQ is of Q ;
becaufe there are as many magnitudes
QQQQQ 1
m <!
.QOOQOJ
Q
Q
as there are in QQQQQ := Q.
The fame demonftration holds in any number of mag-
nitudes, which has here been applied to three.
,*, If any number of magnitudes, &c.
148 BOOK r. PROP. 11. THEOR.
|F the fir ft magnitude be the fame multiple of the
fecondthat the third is of the fourth, and the fifth
the fame multiple of the fecond that the fixth is oj
the fourth, then fiall the firfi, together with the
fifth, be the fame multiple of the fecond that the third, together
•with the fixth, is of the fourth.
Let ^01 9, the firil, be the fame muhiple of ^,
the fecond, that OO 0> ^'^^ third, is of <^, the fourth ;
and let 9 0 0 0, the fifth, be the fame multiple of * ,
the fecond, that OOOOj ^^^ fixth, is of <2>, the
fourth.
Then it is evident, that \ .^ ,^. ,^. .^ ' , the firfl and
fifth together, is the fame multiple of , the fecond,
that \ !r!r^^ k the third and fixth together, is of
looooj
the fame multiple of <2>) the fourth ; becaufe there are as
f #•• 1
many magnitudes i" ] ^^^^ ^ 3P ^s there are
• f 000 \ _ ^
,*, If the firfl: magnitude, &c.
BOOK V. PROP. III. THEOR.
149
F thefirjl of four magnitudes be the fame multiple
of the fecond that the third is of the fourth, and
if any equimultiples whatever of the firjl and third
be taken, thofe Jliall be equimultiples ; one of the
fecond, and the other of the fourth.
The First.
The Second.
Let \ " |- be the iame multiple of
I !
The Third. The Fourth.
which j T T [ is of A ;
take <;^ S S S S > the fame multiple of <
which \ J 3 ? A
[♦♦♦♦
> is of \
♦♦
that ^
Then it is evident,
Tlie Second.
is the fame multiple of |
150
BOOK V. PROP. III. THEOR.
which i
♦♦♦♦
♦ ♦♦♦
♦ ♦♦♦
The Fourth.
- is of ^ ;
J
becaufe <
> contains
> contains
♦♦♦♦'
♦♦♦♦
♦♦♦♦
as many times as
;- contains ■! T^ T^ !> contains ^
♦♦
The fame realbning is applicable in all cafes.
.*. If the firft four, &c.
BOOK V. DEFINITION V.
'51
DEFINITION V.
Four magnitudes, ^^ 01 ^ ^ > ^j are faid to be propor-
tionals when every equimultiple of the firft and third be
taken, and every equimultiple of the fecond and fourth, as.
of the firfl
&c.
of the fecond
of the third ^ ^
♦ ♦♦♦
♦ ♦♦♦♦
♦♦♦♦♦♦
&c.
of the fourtli
&c. &c.
Then taking every pair of equimultiples of the firft and
third, and every pair of equimultiples of the fecond and
fourth,
' — =or^««
= or 31
or ^
= or 33
= or — 1
r
\^<mm cz, =
L
^♦4 C. = °r
♦ ♦ C. = or
then will -
♦ ♦ C, = or
tt C. = or
^♦^ C. = or
152
BOOK F. DEFINITION V.
That is, if twice the firfl be greater, equal, or lefs than
twice the fecond, twice the third will be greater, equal, or
lefs than twice the fourth ; or, if twice the firfl be greater,
equal, or lefs than three times the fecond, twice the third
will be greater, equal, or lefs than three times the fourth,
and so on, as above expreffed.
If
or
or
or
or
or
then
will
&c.
'♦♦♦
♦♦♦
♦♦♦
♦ ♦♦
♦ ♦♦
&c.
c =
6cc.
or Z]
or 31
or ^
or Z]
or ^
&c.
In other terms, if three times the firft be greater, equal,
or lefs than twice the fecond, three times the third will be
greater, equal, or lefs than twice the fourth ; or, if three
times the firft be greater, equal, or lefs than three times the
fecond, then will three times the third be greater, equal, or
lefs than three times the fourth ; or if three times the firft
be greater, equal, or lefs than four times the fecond, then
will three times the third be greater, equal, or lefs than four
times the fourth, and so on. Again,
BOOK V. DEFINITION V.
^Sl
If <
Sec.
then
will
♦ ♦♦♦
♦ ♦♦♦
^ or ^
= or ^
= or Zl
=: or ID
= or ID
= or Zl
= or Zl
= or Z]
= or Zl
= or Zl
&c.
&c.
&;c.
And so on, with any other equimultiples of the four
magnitudes, taken in the fame manner.
Euclid exprefles this definition as follows : —
The firft of four magnitudes is faid to have the fame
ratio to the fecond, which the third has to the fourth,
when any equimultiples whatfoever of the firft and third
being taken, and any equimultiples whatfoever of the
fecond and fourth ; if the multiple of the firft be lefs than
that of the fecond, the multiple of the third is alfo lefs than
that of the fourth ; or, it the multiple of the firft be equal
to that of the fecond, the multiple of the third is alfo equal
to that of the fourth ; or, ii the multiple of the firft be
greater than that of the fecond, the multiple of the third
is alfo greater than that of the fourth.
In future we ftiall exprefs this definition generally, thus :
If M # C = or Zl ''^ ,
when M ^ C = or 313 ;;; ^
154 BOOK V. DEFINITION F.
Then we infer that 0 , the firft, lias the lame ratio
to ^ , the fecond, which ^ , the third, has to ^ the
fourth : exprelTed in the fucceeding demonftrations thus :
# :ii :: 4 : V;
or thus, 0 : It = ^ • V 7
or thus, — ^ =■ : and is read,
" as 0 is to , so is ^ to ^.
And if # : " : : ^ : ip we fhall infer if
M 0 C5 ^= or ^ w , , , then will
M ^ C = or 13 w ^.
That is, if the firfl; be to the fecond, as the third is to the
fourth ; then if M times the firft be greater than, equal to,
or lefs than m times the fecond, then (hall M times the
third be greater than, equal to, or lefs than m times the
fourth, in which M and m are not to be confidered parti-
cular multiples, but every pair of multiples whatever;
nor are fuch marks as 0, ^, , &c. to be confidered
any more than reprefentatives of geometrical magnitudes.
The ftudent fhould thoroughly underftand this definition
before proceeding further.
BOOK V. PROP. IF. THEOR.
^SS
F the firjl of four magnitudes have the fame ratio to
the fecond, which the third has to the fourth, then
any equimultiples whatever of the frji and third
shall have the fame ratio to any equimultiples of
the fecond and fourth ; viz., the equimultiple of the firji Jliall
have the fame ratio to that of the fecond, which the equi-
multiple of the third has to that of the fourth.
m
Let : ■ ::^ :^, then3 :2|::3^:2^,
every equimultiple of 3 and 3 ^ are equimultiples of
^ and ^ , and every equimultiple of 2 ^ and 2 ^ , are
equimultiples of | and ^ (B. 5, pr. 3.)
That is, M times 3 '^ and M times 3 ^ are equimulti-
ples of and ^ , and ;;z times 2 | and w 2 1^ are equi-
multiples of 2 H and 2 ^ ; but • H • • ^ • V
(hyp); .*. if M 3 C =, or ;^ »? 2 |||, then
M 3 ^ C r=, or Z] « 2 ip (def. 5.)
and therefore 3 ^:2||::3^:2^ (def. 5.)
The fame reafoning holds good if any other equimul-
tiple of the firft and third be taken, any other equimultiple
of the fecond and fourth.
,*. If the firfl: four magnitudes, &c.
156
BOOK V. PROP. V. THEOR.
F one magnitude be the fame multiple of another,
which a magnitude taken from thefirfl is of a mag-
nitude taken from the other, the remainder Jhall be
the fame multiple of the remainder, that the whole
is of the whole.
Q
LetQQ
O
= M'^
and
= M-,,
o
C^<^ minus = M' minus M' «>
D
.-. <> =M'(Jminus.),
and /. ^ = M' A.
/, If one magnitude, Sec.
BOOK V. PROP. VI. THEOR. 157
F two magnitudes be equimultiples of two others,
and if equimultiples of thefe be taken from the fir Ji
two, the remainders are either equal to thefe others,
or equimultiples of them.
Q
Let :yQ = M' ■ ; and QQ = M' a ;
o
Q
then 00 minus ni m ::^
o
M' ■ minus /w' « = (M' minus m') b,
and 00 minus m' k := M' a minus /«' 4 :=
(M' minus m') k .
Hence, (M' minus ;«') m and (M' minus tn') k are equi-
multiples of K and k , and equal to * and a 9
when M' minus m' ':^i i.
.'. If two magnitudes be equimultiples, &c.
158
BOOK F. PROP. A. THEOR.
F the firjl of t/ie four magnitudes has the fame ratio
to the fecond which the third has to the fourth,
then if the firjl be greater than the fecond, the
third is alfo greater than the fourth ; and f equal,
equal ; if lefs, lefs.
Let ^ : H : r ip : ^ ; therefore, by the fifth defini-
tion, if %% d ■■, then will ^^ C #4 ;
but if # CZ ■, then ## [Z ■■
and ^fp [=
and .*. ^ C ;► .
Similarly, if ^ ^, or ^ J, then will ^ z^,
or ^ .
.'. If the firfl of four, &c.
DEFINITION XIV.
Geometricians make ufe of the technical term " Inver-
tendo," by inverfion, when there are four proportionals,
and it is inferred, that the fecond is to the firfl as the fourth
to the third.
Let A : B : : C : O, then, by " invertendo" it is inferred
B : A :: I) : C.
BOOK V. PROP. B. THEOR.
'50
F Jour magnitudes are proportionals , they are pro-
portionals alfo when taken inverfely.
Let ^ : O : : ■ : ^ ,
then, inverfely, O : ^ 1 1 : ■ .
If M ^ n « O? then M ■ I] w ^
by the fifth defimtion.
Let M ^ ID /w Q, that is, w Q CZ M ^ ,
.*. M B lU w , or, /« CZ M ■ ;
.*. if w O CZ M ^, then will w C M B
In the fame manner it may be (liown,
that if ;« Q := or Z] M ^ ,
then will m :=, or 13 M B ;
and therefore, by the fifth definition, we infer
that O : ^ : '^ : H .
.', If four magnitudes, &c.
i6o
BOOK V. PROP. C. THEOR.
F the fiyji be the fame multiple of the fecond, or the
fame part of it, that the third is of the fourth ;
the frjl is to the fecond, as the third is to the
fourth.
Let ^ ^ t the firfl:,be the fame multiple of ^, the fecond,
that 7 J, the third, is of ■, the fourth.
♦ ♦.4
♦ ♦
,m 0, M ? ?,>« A
Then _ _
takeMj J
■ ■
becaufe^S is the fame multiple of ^
that J J is of 4 (according to the hypothcfis) ;
and ^M^ is taken the fame multiple ofSS
that M T T is of ? T ,
,*, (according to the third propolition),
M ^ _ is the fame multiple of ^
that M T T is of 4.
BOOK F. PROP. C. THEOR. i6i
Therefore, if M ^ ^ be of ^ a greater multiple than
;// ^ is, then M J i is a greater multiple of ^ than
w A is ; that is, if M S S ^^ greater than m 0, then
M J J will be greater than m ^ ; in the fame manner
it can be fliewn, if M ^ ^ be equal m ^^ then
M J J will be equal m A.
And, generally, if M ^ ^ C =z or ^ //;
then M will be C ^ or ^ ;« ^*
,', by the fifth definition,
■ ■_..♦♦.▲
■ ■•••♦♦•■•
■ ■
Next, let 0 be the fame part of J S
that itk is of T T-
In this cafe alfo 0 : J J :: (ffc : TT.
For, becaufe
■■"*■"-""' WW
is the fame part of ^ ^ that ■ is of ^ ^ ,
1 62 BOOK F. PROP. C. THEOR.
therefore S S is the fame muhiple of
that ^ J is of ^ .
Therefore, by the preceding cafe,
■ ■ . A .. . ^ .
and • ^ • ■■ •• A • ^^
by propofition B.
/. If the firfl be the fame multiple, &c.
BOOK V. PROP. D. THEOR.
163
\^ the fir Jl be to the fecond as the third to the fourth,
and if the firfi be a multiple, or a part of the
fecond; the third is the fame multiple, or the fame
part of the fourth.
and firft, let
be a multiple H;
J J fhall be the fame multiple of ■.
First. Second. Tliird. Fourth.
QQ 00
Take ^^ =r ^
QQ
Whatever multiple
take ^^
then, becaufe
is of I
the fame multiple of 1
• ....♦♦
and of the fecond and fourth, we have taken equimultiples,
^nd )f Y , therefore (B. c. pr. 4),
00
i64 BOOK F. PROP. D. THEOR.
'OCl"^^ . ^^, but (conft.),
and y\y\ is the fame multiple of ■
that ^ is of U .
Next, let B : ^ ^ • ■ V • T J »
and alfo H a part of -^^ ;
then ip fhall be the fame part of J J ,
Inverfely (B. 5.), ^ ' ■ *= ^ J ' V'
but I is a part of ^^ ;
that is, ^ ^ is a multiple of | ;
, by the preceding cafe, X X is the fame multiple of ^
that is, ^ is the fame part of X X
that H is of
,% If the firft be to the fecond, &c.
BOOK V. PROP. VII. THEOR
165
QUAL magnitudes have the fame ratio to the fame
tnagnitiide, and the fame has the fame ratio to equal
magnitudes.
Let ^ = ^ and any other magnitude ;
then 0 : = ♦ •* and : # = : : ^ ,
Becaufe ^ ^ ^,
.-. M # = M ^ ;
/, if M 0 C := or [3 /;/ , then
M ^ C = or ;^ ;« ,
and .*. % : c = ♦ : ■ (B. 5. def. 5).
From the foregoing reafoning it is evident that,
i£ m C > = Of ZD M 0 , then
wHC^orl^ M^
/.■:#=■:♦ (B. 5. def. 5).
,*. Equal magnitudes, &c.
i66 BOOK F. DEFINITION VII.
DEFINITION VII.
WiiKN of the equimultiples of four magnitudes (taken as in
tile fifth definition), the multiple of the firft is greater than
tli:it of the fecond, but the multiple of the third is not
greater than the multiple of the fourth ; then the firft is
laid to have to the fecond a greater ratio than the third
magnitude has to the fourth : and, on the contrary, the
third is faid to have to the fourth a lefs ratio than the firft
has to the fecond.
If, among the equimultiples of four magnitudes, com-
pared as in the fifth definition, we fhould find
• #••# [=■■■■, but
44^44 = '"' ^ W IP V. or if we mould
rnul .iny particular multiple M oi the firft and third, and
a particular multiple m' of the fecond and fourth, fuch,
that M times the firft is C w' times the fecond, but M'
times the third is not CZ w times the fourth, i.e. ^ or
~~1 "; times the fourth ; then the firll is faid to have to
the tiwnd a strcater ratio than the third has to the fourth;
v>r the thial has to the fourth, under fuch circumftances, a
lets ratio than the hrtl has to the feccaid : although feveral
other equimultiples may tend to ibow that the four mag-
nitudes arc piv>portionAls.
This det\i\itivM\ will in tuiure be exprdSbd ti^is : —
ItM fP C rr Q. but M ■ = - Z3 'T ♦ ,
then P : ~ IZ ■ : ♦ .
In the aK>\^ cv' ;- ' ;\ -tSoia* M «ad af aie to be
wnikkitHi jvftrtkn;. cs, at* fike dK iMilli|ilr' M
BOOK F. DEFINITION VII.
167
and m introduced in the fifth definition, which are in that
definition confidered to be every pair of multiples that can
be taken. It muft alfo be here obferved, that ip , U, H ,
and the like fymbols are to be confidered merely the repre-
fentatives of geometrical magnitudes.
In a partial arithmetical way, this may be fet forth as
follows :
Let us take the four numbers, 8 , 7, j c , and
FirJi.
Second.
Third.
Fourth.
8
7
10
Q
lO
14
2C»
24
21
30
^7
32
28
40
36
40
35
50
45
48
42
60
54
56
49
70
63
64
56
80
72
72
63
90
8t
80
70
lOD
".'-
88
77
no
vy
96
84
120
108
104
9'
'3°
117
T12
98
140
126
&c.
&c.
&c
&c.
Among the above multiples we find 16 C 14 and 20
r~ that is, twice the firft is greater than twice the
fecond, and twice the third is greater than twice the fourth ;
and 16^21 and 20 "^ that is, twice the firil is lefs
than three times the fecond, and twice the third is lefs than
three times the fourth ; and among the fame multiples we
can find -: C 56 and V - C that is, 9 times the firft
is greater than 8 times the fecond, and 9 times the third is
greater than 8 times the fourth. Many other equimul-
1 68 BOOK V. DEFINITION VII.
tiples might be selected, which would tend to Ihow that
the numbers %,y, \o, were proportionals, but they are
not, for we can find a multiple of the firlt ^ a multiple of
the fecond, but the fame multiple of the third that has been
taken of the firft not C the fame multiple of the fourth
which has been taken of the fecond ; for inftance, 9 times
the hrll: is C i o times the fecond, but 9 times the third is
not C ^° times the fourth, that is, -: C 70, but 90
not CZ or 8 times the firfl we find C 9 times the
fecond, but 8 times the third is not greater than 9 times
the fourth, that is, O-i-C 63, but Sc is not C When
any fuch multiples as thefe can be found, the hrft (3~)is
faid to have to the fecond (7) a greater ratio than the third
(10) has to the fourth and on the contrary the third
(10) is faid to have to the fourth a lefs ratio than the
firfl (3) has to the fecond (7).
BOOK r. PROP. Fill. THEOR.
109
F unequal magnitudes the greater has a greater
ratio to the fame than the lefs has : and the fame
magnitude has a greater ? atio to the lefs than it
has to the greater.
Let m and be two unequal magnitudes,
and ^ any other.
k
We fliall firft prove that H which is the greater of the
two unequal magnitudes, has a greater ratio to 0 than ,
the lefs, has to ^ ^
that is, ■ : 0 [Z , : # ;
take M' l^/^' #, M' ■, and tn % ;
fuch, that M' ▲ and M' H fhall be each C # ;
alfo take ;;/ ^ the leaft multiple of ^ ,
which will make m
M'
= M'
.*. M' is not CZ f"
butM'
IS
?n
for.
as m' A is the firft multiple which firft becomes C M'^,
than (w minus I ) ^ or;;/ 0 minus ^ isnotCM' JU,
and ^ is not CI M' a,
/. ;;;' 0 minus 0 + # "^"^ be Zl M' Jj + M' A ;
A
that is, ;;;' % mull be i;;^ M' ■ ;
.-. M'
IS
tn
', but it has been fhown above that
170 BOOK F. PROP. Fill. THEOR.
M' m is note »?' # , therefore, by the feventh definition,
m has to 0 a greater ratio than 1:0.
Next we fhall prove that % has a greater ratio to ^ , the
lefs, than it has to j^ , the greater ;
o''# :■ [= • :■•
A
Take /;/ 0, M' ■■, m' #, and M' ||,
the fame as in the firll: cafe, fuch, that
M' A and M' jp will be each C 0 , and m % the leail
multiple of ^ , which firft becomes greater
than M' H = M' || .
.'. m ^ minus ^ is notC M' ^,
and ^ is not CI M' A ; confequently
m % minus # -}- # is ZH M' g -f M' a ;
▲
,*, m' ^ is ID M' ■, and ,'. by the feventh definition,
A
^ has to a| ^ greater ratio than ^ has to ■ .
,'. Of unequal magnitudes, &c.
The contrivance employed in this propofition for finding
among the multiples taken, as in the fifth definition, a mul-
tiple of the firfl greater than the multiple of the fecond, but
the fame multiple of the third which has been taken of the
firft, not greater than the fame multiple of the fourth which
has been taken of the fecond, may be illuftrated numerically
as follows : —
The number 9 has a greater ratio to 7 than has to 7 :
that is, 9 : 7 C : 7 ; or, 8 + i : 7 C = 7-
BOOKF. PROP. Fill. THEOR, 171
The multiple of i , which firft becomes greater than 7,
is 8 times, therefore we may multiply the firft and third
by 8, 9, 10, or any other greater number; in this cafe, let
us multiply the firft and third by 8, and we have '^-^-f- 8
and : again, the firft multiple of ^ which becomes
greater than 64 is 10 times; then, by multiplying the
fecond and fourth by 10, we ftiall have 70 and 70 ; then,
arranging thefe multiples, we have —
8 times lo times 8 times lo times
the first. the second. the third. the fourtli.
6^+ 8 -0 -o
Confequently 04 -j- 8, or 72, is greater than -o, but -t^
is not greater than 70, .•. by the feventh definition, 9 has a
greater ratio to 7 than has to 7 .
The above is merely illuftrative of the foregoing demon-
ftration, for this property could be fhown of thefe or other
numbers very readily in the following manner ; becaufe, if
an antecedent contains its confequent a greater number of
times than another antecedent contains its confequent, or
when a fraction is formed of an antecedent for the nu-
merator, and its confequent for the denominator be greater
than another fraction which is formed of another antece-
dent for the numerator and its confequent for the denomi-
nator, the ratio of the firft antecedent to its confequent is
greater than the ratio of the laft antecedent to its confe-
quent.
Thus, the number 9 has a greater ratio to 7, than 8 has
to 7, for ^ is greater than -.
Again, 17 : 19 is a greater ratio than 13 : 15, becaufe
17 17 X 15 _ 255 J 13 13 X 19 247 ,
evident that ^ is greater than |g, .-. J-^ is greater than
1/2 BOOK F. PROP. VIIT. THEOR.
— , and, according to wliat has been above fliown, 17 has
to 19 a greater ratio than 13 has to 15.
So that the general terms upon which a greater, equal,
or lefs ratio exifVs are as follows : —
A C . .
If g be greater than ^, A is faid to have to B a greater
A C
ratio than C has to D ; if — be equal to rr, then A has to
B the fame ratio which C has to D ; and if -^ be lefs than
^, A is faid to have to B a lefs ratio than C has to D.
The ftudent fhould underftand all up to this propofition
perfectly before proceeding further, in order fully to com-
prehend the following propofitions of this book. We there-
fore ftrongly recommend the learner to commence again,
and read up to this flowly, and carefully reafon at each ftep,
as he proceeds, particularly guarding againlT; the mifchiev-
ous fyftem of depending wholly on the memory. By fol-
lowing thefe inftrudions, he will find that the parts which
ufually prefent confiderable difficulties will prefent no diffi-
culties whatever, in profecuting the ftudy of this important
book.
BOOK V. PROP. IX. THEOR.
^71,
AGNITUDES which have the fame ratio to the
fame magnitude are equal to one another ; and
thofe to which the fame magnitude has t/ie fame
rat to are equal to one another.
Let ^ : ^ : : 0 : p, then ^ = 0 .
For, if not, let ▲ CI 0 ? then will
4 : «^ C # : (B. 5- pr. 8),
which is abfurd according to the hypothefis.
.*. ^ is not C 0 .
In the fame manner it may be fhown, that
A is not ^ ▲,
/. 4 =#.
Again, let H : ^ : : '^' : ^ , then will ^ = 0 .
For (invert.) ^ : || : : f| : H,
therefore, by the firft cafe, A ^ A .
,*. Magnitudes which have the fame ratio, &c.
This may be {hown otherwife, as follows : —
Let ^ : B ^ A : C, then B = C, for, as the fradlion
— = the fradlion -, and the numerator of one equal to the
numerator of the other, therefore the denominator of thefe
fradlions are equal, that is K zz C.
Again, if B : ,\ = C : A , B = C. For, as - = "^,
B muft = (,.
174
BOOK V. PROP. X. THEOR.
HAT magnitude which has a greater ratio than
another has unto the fame magnitude, is the greater
of the two : and that magnitude to which the fame
has a greater ratio than it has unto another mag-
nitude, is the lefs of the two.
Let ^ : C # : ■> then ^ d # .
For if not, let |p =: or ^ 0 ;
then, ^ : si = # : B (^- 5- P^- l) or
^ : H 13 ^ : ■ (B. 5. pr. 8) and (invert.),
which is abfurd according to the hypothefis.
,*, ■ is not =: or ^ ^ , and
.'. S muft he r~ ^.
Again, let «: 0 C V : fP,
then, 0 ID ^>
For if not, 0 mufl: be C or ^ 1^ ,
then flj: 0 Z] p: ^ (B- 5. pr. 8) and (invert.);
or fl: 0 =: H* V (B. 5. pr. 7), which is abfurd (hyp.);
/. 0 is not CZ or = ^ ,
and .'. 0 mufl be ^ ^ .
.*. That magnitude which has, &c.
BOOK V. PROP. XL THEOR.
^75
ATIOS t/iat are the fame to the fame ratio, are the
fame to each other.
Let ^ : ■ = 0 : IP' and 0 : P = ▲ : •,
then will ^ : H = ▲ : •.
For if M ^ C =, or 13 m H,
then M 0 IZ> ^. or 3] w ^,
and if M 0 CZ, ^, or ^ ;/; t' ,
then M A C =, or Z3 m •, (B. 5. def. 5) ;
, if M ^ C, ^, or ^ w d 9 M A [Z, =, or Zl ^« •>
and .*. (B. 5. def. 5) ^ I H ^ A : •.
.*, Ratios that are the fame, &c.
176 BOOK r. PROP. XII. THEOR.
F afijf number of tnagnitiides be proportionals, as
one of the antecedents is to its confeqiient, fo f}:>all
all the antecedents taken together be to all the
confequents.
then will | : # =
■ +0+ +« + ^:# + 0+ +' + ••
For if M U IZ w 0, then M Q [Z w <>,
and M \^tn M • CZ ^« t,
alfoM ▲ IZ'« •• (B. 5. def. 5.)
Therefore, if M JH C w 0, then will
MJ+MQ + M -I-M. + Ma,
or M (H + O + + • + ^) be greater
tlian ;;/ ^ •\- tn ^ -\- m •\' ^'^ ▼ "h ^^^ •>
or;^(# +0+ , + ^ + *)-
In the fame way it may be fhown, if M times one of the
antecedents be equal to or lefs than m times one of the con-
fequents, M times all the antecedents taken together, will
be equal to or lefs than ni times all the confequents taken
together. Therefore, by the fifth definition, as one of the
antecedents is to its confequent, fo are all the antecedents
taken together to all the confequents taken together.
.*, If any number of magnitudes, &c.
BOOK V. PROP. XIII. THEOR.
'77
F the firjl has to the fecond the fame ratio which
the third has to the fourth, but the third to the
fourth a greater ratio than the fifth has to the
fixth ; the firfi fhall afo have to the fecond a greater
ratio than the fifth to the fixth.
Let fP : O = ■ : # , but ■ : A ci O : #,
then fP : D IZ O : ••
For, becaufe | : d O • 0> there are fome mul-
tiples (M' and ;«') of | and ^^ and of ^ and ^,
fuch that M'
m
but M' <^ not C m 0, by the feventh definition.
Let thefe multiples be taken, and take the fame multiples
of ■ and ([n.
/. (B. 5. def. 5.) if M' ^ C, =, or Zl /«' Q ;
then will M' | C =, or ^ m ,
but M' ■ C »^' ^ (conftrudlion) ; ■
.*. M' ^ C ni Q ,
but M' <3 is not C ni 0 (conflrudlion) ;
and therefore by the feventh definition,
W :0 CIO
^^v *
.*. If the firll; has to the fecond, &c.
A A
178
BOOK V. PROP. XIV. THEOR.
F the fir Jl has the fame ratio to the fecondivhich the
third has to the fourth ; then, ifthefirjl be greater
than the third, thefecondjhall be greater than the
fourth; and if equal, equal; andiflefs, lefs.
Let ^ : Q : : B : ^ , and firfl fuppofe
IP [Z » , then will Q C ^ •
For^rQCI : IJ (B.5.pr. 8), andbythe
hypothefis, ^ : O = ^Ji : ^ ;
/. ■ : ♦ CZ :D(B. s-pr-'is).
/. ♦ Z3 D (B- 5- pr- io-)» or Q C ♦•
Secondly, let ^ = |P , then will ^ ^ ^ .
For ^ : O = : D (B. 5. pr. 7),
and ^ : Q = : ^ (hyp.) ;
.*. ■ : D= -V : ♦ (B. 5- pr- lO'
and /. O = 4 (B. 5, pr. 9).
Thirdly, if ^ 13 , then will O ZI ♦ ;
becaufe C W ^"d : ^ = ^ : Q ;
/. ^ C O, by the firft cafe,
that is, Q 13 ^ .
/. If the firft has the fame ratio, &c.
BOOKV. PROP. XV. THEOR. 179
AGNITUDES /lave the fame ratio to one another
which their equimultiples have.
Let 0 and be two magnitudes ;
then, 0 : ■ : : M' 0 : M' ^ ^^
For A : = a
.*. # : H :: 4 • : 4 • (B. 5- pr- 12)-
And as the fame reafoning is generally applicable, we have
# : ■ :: M' A : M'h.
/, Magnitudes have the fame ratio, &c.
i8o BOOKF. DEFINITION XIII.
DEFINITION XIII.
The technical term permutando, or alternando, by permu-
tation or alternately, is ufed when there are four propor-
tionals, and it is inferred that the firft has the fame ratio to
the third which the fecond has to the fourth ; or that the
firft is to the third as the fecond is to the fourth : as is
Ihown in the following propofition : —
Let# : 4 ::19 :B)
by " permutando" or "alternando" it is
inferred ^ : ^ •• ^ • B •
It may be neceffary here to remark that the magnitudes
A, ^j V7H7 muft be homogeneous, that is, of the
fame nature or fimilitude of kind ; we muft therefore, in
fuch cafes, compare lines with lines, furfaces with furfaces,
folids with folids, &c. Hence the ftudent will readily
perceive that a line and a furface, a furface and a folid, or
other heterogenous magnitudes, can never ftand in the re-
lation of antecedent and confequent.
BOOK V. PROP. XVL THEOR.
i8i
F four magnitudes of the fame kind be proportionals,
they are afo proportionals ivhen taken alternately.
Let ^ : Q : : H : ▲ , then ip : B - U • ^ •
ForM fl : M O :: ^ : Q (B. 5. pr. 15),
d M ^ : M Q :: H : ^ (^yP-) ^nd (B. 5. pr. 11)
alfo /;; m : /;; ▲ ' • H * ^ (^- 5- P''- ^ S) >
.*. M ^ : M Q :: w : /« ^ (B. 5. pr. 14),
and /. if M ^ C. =» or ^ zw B ?
then will M Q C :=, or 33 ;« ^ (B. 5. pr. 14) ;
therefore, by the fifth definition,
.*. If four magnitudes of the fame kind, &c.
1 82 BOOK F. DEFIXmOX XFL
DEFLS'ITIOX XVI.
DnmxxDO, by di^ i :- . r - :h ere ire :": _ : r : : : ; - r ,
and it is inferred, l i : J-.e exceli : : : - : toood
b to the fecood, £i iJie ev; ;::;::::; :r : -,
b to tbe fenrth.
le: : 3 ::C : D;
far ** diridendo ** it b inferred
A miners B : B : : C minns '^ : ~" .
Ac; ; r :: . r :: -e,A b fbppt^i :: r-e rti ;' ^ :
B, and C i:'- -" ; if thb be -:: ±: :i : :ut to
have r : i :: ::..- :£ D greater iIjj: .2 :
S :A :-. D :C;
-A :A :: zuz^C :C.
BOOK V. PROP. XVII. THEOR. 183
[F magnitudes, taken jointly, be proportionals, they
Jhall alfo be proportionals ii-hen taken feparately :
\, that is, if tivo magnitudes together have to one of
them the fame ratio which two others have to one
ofthefe, the remaining one of the fir ft two Jhall have to the other
the fame ratio which the remaining one of the laft two has to the
other of thefe.
Let tp + CI: O ::" + ♦: ♦,
then will ^ : O :: ■ : ♦.
Take M ^ C « O to each add M Q,
then we have M V + M Q C 'w O + M Q,
orM(V + CI) C (^^ + M: D:
but becaufe IP + 0:0::"+#: ♦ (hyp.),
and M (IP + O) C (;« + xM) Q ;
.-. M (■ + ♦) C (^^ + M) 4 (B. 5. def. 5) ;
/. M ^ + M ♦[=//;♦+ M ♦ ;
.'. M '^ C ^ ^ . by taking M ^ from both fides :
that is, when 'SI ^ ^ m U, then M T~ m ^ .
In the Tame manner it may be proved, that if
M ^ r= or ^ OT U, then will M =r or — \ m ^ •
and /. V : O : : ? : ♦ (B. 5. def. 5).
.*. If magnitudes taken jointly, &c.
l84 book V. DEFINITION XV.
DEFINITION XV.
The term componendo, by compofition, is ufed when there
are four proportionals ; and it is inferred that the firft toge-
ther with the fecond is to the fecond as the third together
with the fourth is to the fourth.
Let A : B : : : D ;
then, by the term " componendo," it is inferred that
A-|.B:B:: -j-D:D.
By " invertion" B and O may become the firft and third,
A and _ the fecond and fourth, as
B : A : : D : C ,
then, by " componendo," we infer that
B + A : A ; : D -|- . : ^ .
BOOK F. PROP. XVIII. THEOR.
i8s
F magnitudes, taken feparately, be proportionals ,
they fliall alfo be proportionals when taken jointly :
that is, if the Jirji be to the fecond as the third is
to the fourth, the firji. and fecond together fhall be
to the fecond as the third and fourth together is to the fourth.
Let IP : O
then fP + Q : Q
for if not, let |p -f- Q
fuppofing ^
• • ^^ • v^ • •
but ^ : Q : :
not = ^ ;
• (B. 5. pr. 17);
: ^ (hyp.);
.'•■:#::■: 4 (B. 5. pr. n);
•••• = ♦ (B. 5- pr- 9).
which is contrary to the fuppofition ;
.'. ^ is not unequal to ^ ;
that is 0 =: ^ ;
*, If magnitudes, taken feparately, &c.
B B
i86
BOOK V. PROP. XIX. THEOR.
F a isohole magnitude be to a whole, as a magnitude
taken from the firji, is to a magnitude taken from
the other ; the remainder ffoall be to the remainder,
as the ivhole to the whole.
Let l^ + O :■ + ♦:: IP :■,
then will Q: ::'P> + 0:H+'',
For tP + a : V :: ■ + t : ■ (^l^er.),
.*. O : V ••: ♦ :■ (divid.),
again Q : 4 ^^ 9 ^ H (alter.),
butlP + 0:» + # ::^:B hyp.);
therefore Q : : : ^ + D : ■ + ♦
(B. 5. pr. 11).
,*, If a whole magnitude be to a whole, &c.
DEFINITION XVII.
The term " convertendo," by converfion, is made ufe of
by geometricians, when there are four proportionals, and
it is inferred, that the firft is to its excefs above the fecond,
as the third is to its excefs above the fourth. See the fol-
lowing propofition : —
BOOK V. PROP. E. THEOR.
187
F four magnitudes be proportionals, they are alfo
proportionals by converjion : that is, the Jirjl is to
its excefs above the fecond, as the third to its ex-
cefs above the fourth.
then fhall • O • • ^ • ■ '-W,
Becaufe
therefore '
.-. o
•. #0:
:0::B : (divid.),
i :: ^ : ■ (inver.).
(compo.).
.'. If four magnitudes, &c.
DEFINITION XVIII.
" Ex squall " (fc. diflantia), or ex aequo, from equality of
diftance : when there is any number of magnitudes more
than two, and as many others, fuch that they are propor-
tionals when taken two and two of each rank, and it is
inferred that the firft is to the laft of the firft rank of mag-
nitudes, as the firft is to the laft of the others : " of this
there are the two following kinds, which arife from the
different order in which the magnitudes are taken, two
and two."
i88 BOOK V. DEFINITION XIX.
DEFINITION XIX.
" Ex asquali," from equality. This term is ufed iimply by
itfelf, when the firft magnitude is to the fecond of the firft
rank, as the firft to the fecond of the other rank. ; and as
the fecond is to the third of the firft rank, fo is the fecond
to the third of the other ; and fo on in order : and the in-
ference is as mentioned in the preceding definition; whence
this is called ordinate proportion. It is demonftrated in
Book 5. pr. 22.
Thus, if there be two ranks of magnitudes,
A, B, , . , E, F, the firft rank,
and L, M, N , < ' , P, Q, the fecond,
fuch that A : B : : L : M, B : :: M : ,
C : U : : .\ : ( ) , D : E : : o : P, E : F : : P : Q ;
we infer by the term " ex squali" that
A : F :: L :Q.
BOOK F. DEFINITION XX. 189
DEFINITION XX.
" Ex ^quali in proportione perturbata feu inordinata,"
from equality in perturbate, or diforderly proportion. This
term is ufed when the firft magnitude is to the fecond of
the firft rank as the laft but one is to the laft of the fecond
rank ; and as the fecond is to the third of the firft rank, fo
is the laft but two to the laft but one of the fecond rank ;
and as the third is to the fourth of the firft rank, fo is the
third from the laft to the laft but two of the fecond rank ;
and fo on in a crofs order : and the inference is in the i8th
definition. It is demonftrated in B. 5. pr. 23.
Thus, if there be two ranks of magnitudes,
A. , B , C , D , E , F , the firft rank,
and , M , N , O , P , Q , the fecond,
fuch that A : B : : P : Q , B : C : : O : P ,
C^ : D : : N : O , D : ' : : * : N , : : : : vr ;
the term " ex xquali in proportione perturbata feu inordi-
nata" infers that
A : r : : ^ : <,> .
190
BOOK V. PROP. XX. THEOR.
F i/iere be three magnitudes , and other three, which,
taken two and two, have the fame ratio ; then, if
the jirjl be greater than the third, the fourth fiall
be greater than the fixth ; and if equal, equal ;
and if lefs, lefs.
Let ^, 0> J be the firft three magnitudes,
and ^, Oj ^> be the other three,
fuch that fp :0 ::4 :0,andC) :B ::0:#-
Then, if ^ IZ> =» or Z] , then will ^ CI, =,
orZl ^.
From the hypothefis, by alternando, we have
andO :0 ::■:•;
.*. "P :♦::■: • (B. 5- pr- n);
/. if I^F d, =, or Z] , then will ^ C =,
orI3 (B. 5. pr. 14).
,*, If there be three magnitudes, 6cc.
BOOK V. PROP. XXL THEOR.
191
F t/iere be three magnitudes, arid other three which
have the fame ratio, taken two and two, but in a
crofs order ; then if the fir ft magnitude be greater
than the third, the fourth fliall be greater than the
fixth ; and if equal, equal ; and if lefs, lefs.
Let
I, be the firft three magnitudes,
and ^, O*, fpt, the other three,
fuch that ^ : A : : O •# > ^"^ A ' H - • ^ ■ O '
Then, if f C. =. or ID ■, then
will ♦ C =, Zl #.
Firft, let ^ be C ■ :
then, becaufe ^ is any other magnitude,
¥•*!=■' A (6. 5-pr.8);
butO :#::¥: A (^yp-);
.-. O :# !=■ :A (B. 5-pr- 13);
and becaufe {^ : ■ :: ^ : (j (hyp.) ;
and it was fliown that (^ '. % d H ' iil >
.*. O : " C C : ♦ (B. 5- pr- 13);
192 BOOK F. PROP. XXI. THEOR.
•• • =] ♦,
that is ^ C I .
Secondly, let ^ H ; then fhall ^ = ^.
For becaufe ^ B,
V :* = ■ :dl (B. 5.pr.7);
but : il = 0> : (hyp.).
and ^ * A = O : ^ (hyp- ^"'l ifiv.),
.-. O : # = 0 : ♦ (B. 5. pr. II),
.-. ^ = i (B. 5. pr. 9).
Next, let be Z3 ■? then ^ fhall be Z3 ;
for B C
and it has been (hown that (§ • ^ ^ ^ * ▼'
and il : = : O;
/. by the firft cafe is C ^j
that is, ^ ^ 9 .
/. If there be three, &c.
BOOK V. PROP. XXII. THEOR.
193
F there be any number of magnitudes, and as nuuiy
others, 'which, taken two and two in order, have
the fame ratio ; the frji JJjall have to the lajl of
the firft magnitudes the fame ratio which the frji
of the others has to the lajl of the fame .
N.B. — This is ifually cited by the words "ex (egua/i," or
"ex cequo."
irft, let there
36 magnitud
es^
and as many others ▲
,0 =
?
fuch that
w ••
♦ "♦ :
0,
and ^
:il ::0
•
>
then fliall
1^ • ^ • •
▼ •
♦ =
■«.
Let thefe magnitudes, as well as any equimultiples
whatever of the antecedents and confequents of the ratios,
lland as follows : —
and
M ^,« ♦, N ' , M ^, w <;>, N 1,
becaufe |p : ^ : : ^ : O ?
:: M ^ :/«<3 (B. 5. p. 4).
For the fame reafon
w ^ : N : ; /« <^ : N | ;
and becaufe there are three magnitudes,
c c
/.Mm: m
194 BOOK F. PROP. XXII. THEOR.
and other three, M ^ , w <^ , N 0 ,
which, taken two and two, have the fame ratio ;
/. ifMip CZ, =, ori:N B
then will M ^ CZ. =. oi' Z] N , by (B. 5. pr. 20) ;
and .*. ^ : ■ : : ^ : # (def. 5).
Next, let there be four magnitudes, ^ , ^, H ^ ^ »
and other four, ^ , ^, IB , ▲ ,
which, taken two and two, have the fame ratio,
that is to fay, ^ • ^ • • O ' #'
♦ :■::•: ,
and A : ^ ::m : ▲,
then fhall IP : ^ : : O * ^ '
for, becaufe l[p , ^^, , are tliree magnitudes,
and <^ , ^f , other three,
which, taken two and two, have the fame ratio ;
therefore, by the foregoing cafe, ^^ : ■ : : (2> • ^,
but a : 4 : • «■ : -^ ;
therefore again, by the firfl; cafe, ip : ^ : : (^ '- ^ f
and fo on, whatever the number of magnitudes be.
,*, If there be any number, &c.
BOOK V. PROP. XXIII. THEOR.
195
F t/iere be afiy number of tnagnitudes, and as many
others, ivhich, taken two and two in a crofs order,
have the fame ratio ; the firjl fliall have to the laji
of the firjl magnitudes the fame ratio which the
firji of the others has to the laji of the fame.
N.B. — This is ifually cited by the words " ^x aquali in
proportione perturbatd ;" or " ex aquo perturbato."
Firft, let there be three magnitudes, ^j(^> |)
and other three, ' > O ' ^ »
which, taken two and two in a crofs order,
have the fame ratio ;
o
Let thefe magnitudes and their refpective equimuhiples
be arranged as follows : —
M ,M^,m^,M ,,,m(^,m%,
then f IQ ::M ' : M Q (B. 5. pr. 15);
and for the fame reafon
but^ :q ::<2> :0 (hyp.).
that is, |; :
U '
:o
and Q
:■ :
•♦
then fhall ^
:■ :
= ♦
196
BOOK V. PROP. XXIII. THEOR.
.-. M ip :MQ ::<^ :# (B. 5. pr. n);
and becaufe O : ■ : : ^ : <2> (Jiyp-)>
.-. M Q : w H : : ^ : w ^ (B. 5. pr. 4) ;
then, becaufe there are three magnitudes,
M W, M Q, w ■,
and other three, M , m (2), w ^ ,
which, taken two and two in a crofs order, have
the fame ratio ;
therefore, if M [^, ^, or "H ;;; J j
then will M [Z, =r, or ;i] ;;/ 0 (B. 5. pr. 21),
and /. ,;; : ■ :: -J. : # (B. 5. def. 5).
Next, let there be four magnitudes,
and other four, (2)j ^j ■> A.?
which, when taken two and two in a crofs order, have
the fame ratio ; namely.
IP
:D
:: ■
D
■
::•
andH
• •#
-0
en fhall
"O
For, becaufe ^^ ^, | are three magnitudes,
BOOKF. PROP. XXIII. THEOR. 197
and 9, SI, i^, other three,
which, taken two and two in a crofs order, have
the fiime ratio,
therefore, by the firfl cafe, ^ : H •• 0 • ^^
but ■ : :: <^ : #,
therefore again, by the firft cafe, y : ^ : : /S ' A ?
and i'o on, whatever be the number of fuch magnitudes.
.*. If there be any number, &c.
198
BOOK V. PROP. XXIV. THEOR.
jF the firji has to the fecond the fame ratio which
the third has to the fourth, and the fifth to the
fecond the fame which the fix th has to the fourth,
the fir fi and fifth together Jhall have to the fecond
the fame ratio which the third and fix th together have to the
fourth.
First.
Fifth.
Second.
D
Third.
Sixth.
Fourth.
Let ip :
U:
:a:<^,
and (2> :
D:
:•:#.
'+0
•Q:
• ■ + • : 4
then
For <2>:D--: #: ^ (%P-).
and Q : ^ :: ^ : B (^yP-) ^"^ (invert.),
.-. 0> •¥::#:■ (B- 5- Pr- 22);
and, becaufe thefe magnitudes are proportionals, they are
proportionals when taken jointly,
.•• V+ 0:0:: •+ ■: • (B. 5- pr- 18),
but o : D • : • • '- (hypO.
.-. V + O : U ::#+■• t (B. 5- pr. 22).
/. If the firft, &c.
BOOK V. PROP. XXV. THEOR.
199
F four magnitudes of the fame kind are propor-
tionals, the greatejl and leaf of them together are
greater than the other two together.
Let four magnitudes, ■ -j- ^, H -|- ■-' , |^, and |^ ,
of the fame kind, be proportionals, that is to fay,
and let ■ -f- O ^^ ^^ greateft of the four, and confe-
quently by pr. A and 14 of Book 5, ^ is the leaft ;
then will ^+1314- beClB+ +D;
becaufe If + Q :■+>:: O : ♦,
but
+ Dl= ■ +
(B. 5. pr. 19),
(hyp.).
.'. "f [= ■(B. 5. pr. A);
to each of thefe add O "4" ^7
•*. fP + O + 1= ■ + o + ♦■
If four magnitudes, &c.
2o,o BOOK V. DEFINITION X.
DEFINITION X.
When three magnitudes are proportionals, the firfl is laid
to have to the third the dupHcate ratio of that which it has
to the fecond.
For example, if A, b', C, be continued proportionals,
that is, A : B :: B : C, A is faid to have to C the dupli-
cate ratio of x\ : B ;
or — r= the fquare of — .
This property will be more readily feen of the quantities
'J ^"f , , J, tor /T !' '. u ' '.'. li ■ '• a \
and — ^ r^ r= the fquare of — = r.
or of iJy
f jr~ ,
for — ^ -3 = the fquare of — =:— .
a r " '
DEFINITION XI.
When four magnitudes are continual proportionals, the
firft is faid to have to the fourth the triplicate ratio of that
which it has to the fecond ; and fo on, quadruplicate, &c.
increafing the denomination ftill by unity, in any number
of proportionals.
For example, let. A, B, C, D, be four continued propor-
tionals, that is, A ; : : : : C :: C : D ; A is faid to have
to D, the triplicate ratio of N to iJ ;
or - := the cube of—.
BOOK K DEFINITION XL 201
This definition will be better underftood, and applied to
a greater number of magnitudes than four that are con-
tinued proportionals, as follows : —
Let^r", ' yar> ^y be four magnitudes in continued pro-
portion, that is, ^ »■':': : '■ ar '-'-ar '• (i,
. ar' „ , , -ar^
then =: r" r= the cube or — ^ r.
a
Or, let ar', ar*, ar^, ur', ar, a, be fix magnitudes in pro-
portion, that is
ar* : rtr* :: ar^ ■ ar* :: ar" : ar" :: ar' : ar :: ar : a,
a r - a r
then the ratio — = r" zrz the fifth power of — : zr: r.
a ^ rtr*
Or, let a, ar, ar^, ar^, ar*, be five magnitudes in continued
proportion; then — 5 := -5 =z the fourth power of — ::=:-.
DEFINITION A.
To know a compound ratio : —
When there are any number of magnitudes of the fame
kind, the firfi: is faid to have to the lafl: of them the ratio
compounded of the ratio which the firfl has to the fecond,
and of the ratio which the fecond has to the third, and of
the ratio which the third has to the fourth ; and fo on, unto
the lafl; magnitude.
For example, if A , B , C , D ,
be four magnitudes of the fame
kind, the firft A is faid to have to
the lafl: D the ratio compounded
of the ratio of A to B , and of the
ratio of B to C , and of the ratio of C to D ; or, the ratio of
DD
A
B
C
D
E
F
G
H
s
K
L
202 BOOKF. DEFINITION A.
A to D is faid to be compounded of the ratios of \ to B ,
B to C , and c to |j.
And if A has to B the fame ratio which 1 has to V , and
B to C the fame ratio that G has to H, and C to D the
fame that K has to L ; then by this definition, \ is said to
have to L> the ratio compounded of ratios which are the
fame with the ratios of E to F, G to H, and K to L. And
the fame thing is to be underftood when it is more briefly
exprefled by faying, \ has to D the ratio compounded of
the ratios oft to F, G to H, and K to I .
In like manner, the fame things being fuppofed ; if
has to the fame ratio which \ has to D, then for fhort-
nefs fake, is faid to have to the ratio compounded of
the ratios of E to F, G to H, and K to L.
This definition may be better underftood from an arith-
metical or algebraical illuftration ; for, in fact, a ratio com-
pounded of feveral other ratios, is nothing more than a
ratio which has for its antecedent the continued produdl of
all the antecedents of the ratios compounded, and for its
confequent the continued produdl of all the confequents of
the ratios compounded.
Thus, the ratio compounded of the ratios of
2 : ;, 4 : 7, 6 : 1 1, 2 : 5,
is the ratio of ; X X 6 X 2 : X X 1 1 X 5,
or the ratio of 96 : 11 55, or -^2 : 385.
And of the magnitudes A, B, C, D, E, F, of the fame
kind, A : F is the ratio compounded of the ratios of
A : B, B : C C : D, D : E, E : F ;
for A X B X X X E : B X C X x E X F,
^^ nx'x XEXF = T' ""^ ^^^ ""^"^ °^ "^ '■ ^'
BOOK r. PROP. F. THEOR.
203
ATIOS wAic/i are cojnpounded of the fame ratios
are the fame to one another.
Let A : B : : F : G,
B : C :: G : H,
C: D::H:K,
and D : E :: K : L.
A B C D E
F G H K L
Then the ratio which is compounded of the ratios of
A : R, ^ : , : , : t , or the ratio of A : E, is the
fame as the ratio compounded of the ratios of F : G,
G : H, H : K, K : L, or the ratio of F : L.
For ^ =
F
G'
B
C ~"
G
H'
C __
D "■
H
K'
a„d^ =
K
AX
XX
F X
X
X
X
X -:
X
X X ■ —
X L
and /. -
F
— L
or the ratio of A : E is the fame as the ratio of F : L.
The fame may be demonflrated of any number of ratios
fo circumftanced.
Next, let A : B : : K : L,
B: C:: H: K,
C: D:: G: H,
D: E :: F: G.
204 BOOK V. PROP. F. THEOR.
Then the ratio which is compounded of the ratios of
A : B, B : C, C : D, D : E, or the ratio of A : E, is the
fame as the ratio compounded of the ratios of :L, : K,
G : H, F : , or the ratio of F :L.
For - = -,
I
and — =: — ;
r.
A X X X D . X X X F
X ^^ X E — L X X X G *
^•■"^ •••! = -'
F
L
or the ratio of A : ¥ is the fame as the ratio of F : L.
,", Ratios which are compounded, &c.
BOOK V. PROP. G. THEOR.
205
F fever al ratios be the fame to fever al ratios, each
to each, the ratio which is compounded of ratios
which are the fame to the firft ratios, each to each,
jhall be the fame to the ratio compounded of ratios
which are the fame to the other ratios, each to each.
A B C: D E ¥ G H
P Q R S T
a bed e f g h
V w X y
If A : B : : d : ^
and A : B : : P :
Q
a:b::
: \\
CD ::€ -.d
C:D::Q:
R
c:d::
w
: X
E:F ::e:f
E:F ::R
S
e:f::
X
: Y
and G : II :: g : A
G:H:: S :
T
g:h::
Y
: Z
then P : T = ^ "
• •
p^^ P A a
Z3
>
2 — ^' - i-
R D d
=
>
R __ E e
S" — * F — 7
^
9
^ G ff
f H h
)
and • '' X 9 X k X ■ __
^""^ • • 0 X R X s X r —
X
X
X X
X X
~ »
and /. -p = -
~ >
01
rP : T = :
'/..
If feveral ratios, &c.
2o6
BOOK V. PROP. H. THEOR.
F a ratio which is compounded of fever al ratios be
the fame to a ratio which is compounded of fever al
other ratios ; and if one of the firjl ratios, or the
ratio which is compounded of fever al of them, be
the fame to one of the laji ratios, or to the ratio which is com-
pounded of Jeveral of them ; then the remaining ratio ofthefirjl,
or, if there be more than one, the ratio compounded of the re-
maining ratios, JJi all be the fame to the remaining ratio of the
la/i, or, if there be more than one, to the ratio compounded of thefe
remaining ratios.
A
B
C
D
E
F
G
H
F
Q
R
S
T
X
Let A : B, B : C, C : D, D : E, E : F, F : G, G : H,
be the firft ratios, and P : Q^_Qj^R, R : S, S : T, T : X,
the other ratios ; alfo, let A : H, which is compounded of
the iirfl: ratios, be the fame as the ratio of P : X, which is
the ratio compounded of the other ratios ; and, let the
ratio of A : E, which is compounded of the ratios of A : B,
B : C, C : D, D : E, be the fame as the ratio of P : R,
which is compounded of the ratios P : Q,^ Qj R.
Then the ratio which is compounded of the remaining
firft ratios, that is, the ratio compounded of the ratios
E : F, F : G, G : H, that is, the ratio of E : H, fhall be
the fame as the ratio of R : X, which is compounded of
the ratios of R : S, S : T, T : X, the remaining other
ratios.
Becaufe -
BOOK V. PROP. H. THEOR. 207
, X f. X C X D X L X J: X & P X Q X R X S X I
l; X L X D X E X F X G X H Q X R X s X 'I' X X'
, X [; X C X 1^ w K X F X G P X Q w R X S x f
f! X L X D X K •^ h- X(,XU — O X R ^ ^ X T X X'
anH ^ X B X C X D _ P X Q
^ E XCX DX E — Qx R'
, E X F X G R X - X I
• * F X G X H i X I X X»
/. E : H = R : X.
,*, If a ratio which, &c.
2o8
BOOK V. PROP. K. THEOR.
F t/iere be any number of ratios, and any number of
other ratios, fuch that the ratio which is com-
pounded of ratios, which are the fame to the frji
ratios, each to each, is the fame to the ratio which
is compounded of ratios, which are the fame, each to each, to
the lajl ratios — and if one of the firji ratios, or the ratio which
is compounded of ratios, which are the fame to federal of the
firjl ratios, each to each, be the fame to one of the lajl ratios,
or to the ratio which is compounded of ratios, which are the
fame, each to each, to fever al of the lajl ratios — then the re-
maining ratio of the firjl ; or, if there be more than one, the
ratio which is compounded of ratios, which are the fame, each
to each, to the remaining ratios of the firJi, Jhall be the fame
to the remaining ratio of the lajl ; or, if there be more than
one, to the ratio which is compounded of ratios, which are the
fame, each to each, to thefe remaining ratios.
h k m n s
AB, CD, EF, GH, K L, MN.
a b c d e t g
O P , O R , S T , V W , X Y ,
h k I m n p
abed e i g
Let A:B, C:D, E:F, G:H, K:L, M:N, be the
firft ratios, and o :!', (^:R , ^ :T, V :W, X : , the
other ratios ;
and let A : B
zn a '. b ,
C :D
= b :c.
E :F
HZ L id.
G :H
~~7. 'i : '' J
K : L
— e :f.
M:N
T"^ / I P" •
BOOK F. PROP. K. THEOR. 209
Then, by the definition of a compound ratio, the ratio
of ,7 In- is compounded of the ratios of j :/,, /; ic, c -Jt J ','>
g :/"./"■?-. which are the fame as the ratio of A : B, C : D,
E : F, G : H, K : L, M : N, each to each.
Alfo,
:^ ^; h
•k.
Q^
\R •=. k
:/.
: :
T = /:
m.
V :
VV = m
: n.
-- ;
— n
P-
Then will the ratio oi h\p be the ratio compounded of
the ratios of h:k, k:l, I '.my m'.n, n:p, which are the
fame as the ratios of :p , Ct :R , S :T , V :W , X :Y ,
each to each.
/, by the hypothefis a '•!? = h:p.
Alfo, let the ratio which is compounded of the ratios of
A: B, C : D, two of the firfl: ratios (or the ratios of j ict
for \ : ^ = J : A, and C" : P = , : ), be the fame as the
ratio of a : d, which is compounded of the ratios of a : b,
b : c, c : d, which are the fame as the ratios of : ,
: , : , three of the other ratios.
And let the ratios of h : s, which is compounded of the
ratios of h : k, k : m, m : n, n : s, which are the fame as
the remaining firft ratios, namely, E : F, G : H, K : L,
M : N ; alfo, let the ratio of e : g, be that which is com-
pounded of the ratios e : f, f : g, which are the fame, each
to each, to the remaining other ratios, namely, V :W,
X : Y . Then the ratio of h : s fhall be the fame as the
ratio of e : g ; or h : s r= erg.
p AXCXKX'.XKXM g X 6 X ,- X ,i X r X /•
:-^ X 11 X F X H X I X X — - 6 X c X i X . X/ X y '
£ £
2IO BOOK V. PROP. K. THEOR.
, OX ox ^X . X ■ ^x^x^x?»x»
^ X R X I X V, X kXlXmXnXp
by the compolition of the ratios ;
. cX/XcXrfXfX; h X kX I XrnXn
''iXcXdXfXfXj kX I Xm X n Xp
(hyp.).
uXl w c X ^ X £ X / kX kX I w
.7 X c ^ dX cX/Xg kX t Xm ^ n Xp'
but — — A X C __ ^- X X . a Xb Xc __ h Xk X i .
;X. £XD ;X X bxcxd kX I Xm'
, f X -J X f X ; _. »t X n
' ' i X e X fX i' n Xp'
A„J C X ^" X t X • h X k X m X n ,, .
And .v. V^vV — k VmVn Vs ("7?-).
>?Xc X/X^
k
X m
X
n X s
and
m X n
n Xp
—
e
T
X f
Xg
(hyp.).
•
• •
h X k X m
k X m X n
X
X
n .
s
—
e f
•
• •
s
^^
e
/. h
: s :
—
e :
g-
/, If there be any number, &c.
; * Al"-ebraical and Arithmetical expositions of the Fifth Book of Euclid are given m
Hyrne's Doctrine of Proportion ; published by Williams and Co. London. 1841.
BOOK VI.
DEFINITIONS.
I.
ECTILINEAR
figures are faid to
be fimilar, when
they have their fe-
veral angles equal, each to each,
and the fides about the equal
angles proportional.
II.
Two fides of one figure are faid to be reciprocally propor-
tional to two fides of another figure when one of the fides
of the firft is to the fecond, as the remaining fide of the
fecond is to the remaining fide of the firft.
III.
A STRAIGHT line is faid to be cut in extreme and mean
ratio, when the whole is to the greater fegment, as the
greater fegment is to the lefs.
IV.
The altitude of any figure is the straight line drawn from
its vertex perpendicular to its bafe, or the bafe produced.
2;2
BOOK VI. PROP. I. THEOR.
RIANGLES
and parallelo-
grams having the
fame altitude are
to one another as their bafes.
Let the triangles
1 and m
have a common vertex, and
their bafes
and
in the fame ftraight hne.
Produce i both ways, take fucceffively on
— — produced lines equal to it ; and on — — — pro-
duced lines succefTively equal to it ; and draw lines from
the common vertex to their extremities.
A
The triangles j^-JKJt^ thus formed are all equal
to one another, fmce their bafes are equal. (B. i . pr. 38.)
A
and its bafe are refpectively equi-
i
multiples of ■ and the bafe
BOOK VL PROP. I. THEOR. 2- 3
^
In like manner » _ and its bafe are refpec-
i
lively equimultiples of |^ and the bafe — — .
.*. Ifm or 6 times ^ (^ := or 13 « or 5 times B
then m or 6 times — — C ^ or ;^ « or 5 times u-mi ,
m and « ftand for every multiple taken as in the fifth
definition of the Fifth Book. Although we have only
fhown that this property exifts when m equal 6, and n
equal 5, yet it is evident that the property holds good for
every multiple value that may be given to m, and to n.
a
(B. 5. def 5.)
Parallelograms having the fame altitude are the doubles
of the triangles, on their bafes, and are proportional to
them (Part i), and hence their doubles, the parallelograms,
are as their bafes. (B. 5. pr. 15.)
Q. E. D.
214
BOOK VI. PROP. II. THEOR.
*
F a Jlraight line
be draivn parallel to any
Jide ■——■■-> of a tri-
angle, it fliall cut the other
tides, or thoj'e Jides produced, into pro-
portional fegments .
And if any Jlraight line ^— ^—
divide the fides of a triangle, or thofe
fides produced, into proportional feg-
ments, it is parallel to the remaining
fide —■•■-■■■■■»,
Let
PART I.
I, then {hall
• •• ' ■••
Draw
and
and
(B. I. pr. 37);
■V-
V
\-
^ : ! \ (B.5.pr.7);but
(B. 6. pr. 1),
(■■■■■■■■■* • I
• «■■■■«■■■« * IttlllllBlB*,
(B. 5. pr. II).
BOOK VI. PROP. 11. THEOR.
21
PART II.
Let
Let the fame conftrudlion remain.
becaulc
: :: 1/ : • \
> (B.6. pr. I);
and
■ »«aiaflVB»«
■ «
• • •
but
«■■■ ^aai^
i\
/
-Z=
(hyp.),
), : i \ (B. 5. pr. 1 1 .)
:. (B. 5- pr- 9) ;
■■■■■■•. , and at the
but they are on the fame bafe -■■■■■•■i
fame fide of it, and
•°. II (B- i.pr. 39)
Q. E. D.
2l6
BOOK VI. PROP. III. THEOR.
RIGHT /ine ( )
bifeSling the angle of a
triangle, divides the op-
pofite Jide into fegments
— — ™) proportional
to the conterminous Jides (-
)•
And if a Jiraight line (-
— )
drawn from any angle of a triangle
divide the oppojite Jide ( ■■■■■■)
into fegments ( , ....■■■■..)
proportional to the conterminous fides (— — — , ■ ),
it bifeSls the angle.
PART I.
Draw -■
to meet
then, ^ ^ (B. i.pr. 29),
^ = < ; but ^ = 1 , .-. ^ =-# ,
.*. .......... = I I (B. I. pr. 6);
and becaufe
■ ■■■■■■■IHB
(B. 6. pr. 2) ;
but
(B. 5. pr. 7).
BOOK FI. PROP. III. THEOR. 217
PART II.
Let the fame conftrudtion remain,
and
(B. 6. pr. 2) ;
but — ^.^ : ......... :: «-..—. ; ...^^ (hyp.)
■■.■■■•
(B. 5. pr. .1).
and ,*, ■.■■•■■■a. zzz ■ (B. c. pr. o),
and /. ^ = ^(B. I. pr. 5); but fince
II «..; ^ _ ^^
and ^ = ^ (B. i. pr. 29);
/. ^ =y, and =: ^,
and .*. ........ bifedls ^
Q^E. D.
F F
«^i8
BOOK VI. PROP. IV. THEOR.
N equiatigular tri-
angles ( ^ \
and ,-•'* \ ) the fides
about the equal angles are pro-
portional, and the Jides which are
^L oppojite to the equal angles are
homologous.
Let the equiangular triangles be fo placed that two fides
^^ and
oppofite to equal angles
^^^ may be conterminous, and in the fame ftraight line;
and that the triangles lying at the fame fide of that flraight
line, may have the equal angles not conterminous,
ofite to jtKk , and fl^ to j^|^ .
1. e.
opp
Draw -■•■■••■••t and
', Then, becaufe
▲ = ▲
II
and for a like reafon, •■•■—■•■•
- - „ /
;B.i.pr.28);
■ ""•"5
is a parallelogram.
But
(B. 6. pr. 2) ;
BOOK FI. PROP. IF. THEOR. 219
and lince ' ^ — — — (B. i. pr. 34),
^_i. : .......«i.. * and by
• •
alternation, — ^— ^ : — — — ^ j: -—.-««— ; —
(B. 5. pr. 16).
In like manner it may be fhown, that
■^i^HMiMaBH^ ■«»•«■«■«»« ^^B^i^ia^^^ iiMiJiiaiiia*
and by alternation, that
•■■^^■^■Ma o^^^^^^^ JJ •■■■■■■•■^ J ■■■■■■■■■■*
but it has been already proved that
^a^^^^^^^m * ^MMHHWHHMM * ■■■■■■■■■■«
and therefore, ex squali.
■■■■■■■■■« * ■■■■■■■■■■■
(B. 5. pr. 22),
therefore the fides about the equal angles are proportional,
and thofe which are oppolite to the equal angles
are homologous.
Q. E. D.
220 BOOK VI. PROP. V. THEOR.
F tivo triangles have their Jides propor-
tional (•■■•■•■■- : ■■«.■■■■»
:: ^mmmmmmm, \ ) and
('
■■■■■■■■■■■» * «•■■■■«■«
:: — — ^— : i^^»— .) they are equiangular,
and the equal angles are fubtended by the homolo-
gous fides.
From the extremities of
, draw
and
, making
W= iB (B. i.pr. 2.1;);
= (B. I. pr. 32) ,
and fince the triangles are equiangular.
(B. 6. pr. 4);
but
(hyp.);
and confequently
(B. 5. pr. 9).
In the like manner it may be fhown that
BOOK VI. PROP. V. THEOR. 221
Therefore, the two triangles having a common bafe
— «^— , and their fides equal, have alfo equal angles op-
A =^a„d^ = ^
polite to equal fides, i. e.
(B. I. pr. 8).
But ^F = j^^ (conft.)
and .*• jj^^ =: M/^ ^ for the fame
reafon ^^^ z= flU? ^"^
confequently ^^ := (B. i. 32);
and therefore the triangles are equiangular, and it is evi-
dent that the homologous fides fubtend the equal angles.
CUE. D.
222
BOOK VI. PROP. VI. THEOR.
4
.A
F tivo triangles ( _^**^ _^*'^
and .^^___^ ) have one
angle ( wKk ) o/' ///<' one, equal to one
, angle ( m \) of the other, and the Jides
^ about the equal angles proportional, the
HPI^ triangles Jhall be equiangular, and have
thofe angles equal which the homologous
Jides fubt end.
From the extremities of
of
Z:^
about
^— , one of the fides
■ \ , draw
and
maki
in
g
▼ =A
and ^W zz.
^=4
then ^ =:
(B. I. pr. 32), and two triangles being equiangular.
■ ■^■■■•■■■a
(B. 6. pr. 4) ;
but •••■>•••••••
(hyp.);
(B. 5. pr. 11),
and confequently
•«■■«■«•■•*
(B. 5. pr. 9);
k
BOOK VI. PROP. VI. THEOR. 223
.*. -^ \ = >,♦* in every refpedl.
(B. I. pr. 4).
But ^yf = j^ (conft.),
and /. ZLj ■=. J^ ; and
fince alio ■ \ ::z: JHl ,
^\ = -^ (B. i.pr. 32);
j/\
and .*. A*I,....dW ^"d -^ \ are equiangular, with
their equal angles oppolite to homologous Tides.
Q^E. D.
224.
BOOK VI. PROP. VII. THEOR.
A
/\
F two triangles (
A
and
A
»
* * ) Aave one angle in
• each equal ( ' equal to ^^ ), the
\ Jides about two other angles proportional
\ (-^ : — - :: : ..-—),
^l
A
and each of the remaining angles (
and ^..^ ) either lefs or not lefs than a
right angle, the triangles are equiangular, and thofe angles
are equal about which the Jides are proportional.
Firft let it be alTumed that the angles ^^ and <.^
are each lefs than a right angle : then if it be fuppofed
that i^A ^"'^ ^^ contained by the proportional fides,
are not equal, let ^\ be the greater, and make
Becaufe ^ = ^ (hyp.), and ^\ = ^J (conft.)
/. ^», = ^^--B ^B. I. pr. 32);
\
BOOK VI. PROP. FIT. THEOR. 225
(B. 6. pr. 4),
but — ^^— : ■ :: — — ■■— : — .— .. (hyp.)
• 9
(B. 5. pr. 9),
4
and ,*. ^^ = ^^ (B. 1. pr. 5).
But ^^B is lefs than a right angle (hyp.)
,•, ^^^ is lefs than a right angle ; and ,', ^B muft
be greater than a right angle (B. i. pr. 13), but it has been
proved ^ '^^^.^ and therefore lefs than a right angle,
which is abfurd. ,*. ^<^ and ^-\ are not unequal ;
.', they are equal, and fince ^B rz / \ (hyp.)
4=4
(B. I. pr. 32), and therefore the tri-
angles are equiangular.
-^ and ^5
But if '^^ and ^*-^ be aflumed to be each not lefs
than a right angle, it may be proved as before, that the
triangles are equiangular, and have the fides about the
equal angles proportional. (B. 6. pr. 4).
Q. E. D.
OG
226
BOOK VI. PROP. Fin. THEOR.
N a right angled
triangle
.j^S^ 9 ^^
(
triangle and to each other.
Becaufe ^^p» ~
common to
i
a perpendicular (
be drawn from the right angle
to the oppojitejide, the triangles
) on each Jide of it are fimilar to the whole
(B. I. ax. 1 1 ), and
and
t\^<
;B. I. pr. 32);
and ..^^^^l are equiangular ; and
conlequently have their Tides about the equal angles pro-
portional (B. 6. pr. 4), and are therefore limilar (B. 6.
def. I).
In like manner it may be proved that ^^ is fimilar to
k
; but
has been ihewn to be limilar
to
and
k
are
fimilar to the whole and to each other.
Q. E. D.
BOOK VI. PROP. IX. PROB.
22:
ROM a given Jiraig/it line { " '" )
to cut off any required part .
From either extremity of the
given line draw — ^■—"••••■t. making any
angle with ■ ; and produce
■•••••> till the whole produced line
■mtiBH* contains ■ > as often as
-■■"■"■- contains the required part.
Draw , and draw
■ is the required part of
For fi
nee
* ■«■■•••«
(B. 6. pr. 2), and by compolition (B. 5. pr. 18) ;
— ^— > -■".- mmmmmm
but mmm
'•■--•■ contains
as often
as
contains the required part (conft.) ;
■■■— is the required part.
Q. E. D.
228
BOOK VI. PROP. X. PROB.
and
draw
O divide a Jlraight
line ( )
fanilarly to a
given divided line
)•
From either extremity of
the given line — i^
draw ■■■■««a>s»aaaKaj»«M
making any angle ; take
and
>■•••«« equal to
refpedlively (B. i. pr. 2) ;
and draw — — --— and
-— II to it.
or
and
Since
( — j are II,
(B.6. pr. 2),
(B. 6. pr. 2),
and ,*, the given line
fimilarlv to
(conft.),
(conft.).
is divided
Q.E. D.
BOOK VL PROP. XL PROB.
229
O yf«i/ a third proportional
to two given Jlraight lines
At either extremity of the given
line ^— i^— » draw .---——
making an angle ; take
....... .^ = , and
draw I :
make .,.._... =: ,
and draw || — ^ •
(B. I. pr. 31.)
lu - is the third proportional
to -■^— ^^ and _
For fince
but
(B. 6pr. 2);
■(conft.) ;
(B. 5. pr. 7).
Q^E. D.
230
BOOK VI. PROP. XII. PROB.
O find a fourth pro-
portional to three
given lines
Draw
and
take
and
alfo
draw
and
making any angle ;
(B. r. pr. 31);
is the fourth proportional.
bU. (;■.
On account of the parallels,
(B. 6. pr. 2);
•} = { =
.} (conft.);
■ ■■•■■■■•« •
(B. 5. pr. 7).
Q^E. D.
BOOK VI. PROP. XIII. PROP.
^31
O Jind a mean propor-
tional between two given
Jlraight lines
{
«ia«MMl«a«ni«l
}
Draw any ftraight line
make —
and
bifed
and from the point of bifedtion as a centre, and half the
line as a radius, defcribe a femicircle
draw — ^— ^— JL — — —
cs
is the mean proportional required.
Draw
and
Since '^^^ is a right angle (B. 3. pr. 31),
and ^^^^— is J_ from it upon the oppofite fide,
•*. •^^"■^ is a mean proportional between
— and ' (B. 6. pr. 8),
and .*. between — — — and — •• (conft.).
Q. E. D
232 BOOK VI. PROP. XIV. THEOR.
QJJ A L parallelograms
\
and
•which have one angle in each equal,
have the Jides about the equal angles
reciprocally proportional
(
■)•
II.
And parallelograms which have one angle in each equal,
and the fides about them reciprocally proportional, are equal.
Let
and
- and
; and
and
^■^~", be fo placed that '■■■ ' ■
-■— may be continued right Unes. It is evi-
dent that they mayaflume this pofition. (B. i. prs. 13, 14,
1 5-)
Complete
%
Since
•V
\ \ \
(B. 5. pr. 7.)
BOOK VI. PROP. Xir. THEOR. 233
(B. 6. pr. I.)
The fame conftrudtion remaining :
r
A
(B. 6. pr. I.)
— (hyp.)
(B. 6. pr. I.)
(B. 5. pr. II.)
and .*. ^Hi^ = ^^ (B. 5. pr. 9).
Q^E. D.
H H
234
BOOK VI. PROP. XV. THEOR.
I.
QUAL triangles, which have
one angle in each equal
( ^^ ^ ^B ), have the
JiJt's about the equal angles reciprocally
proportional
(■
-- )•
II.
j^i
And two triangles which have an angle of the one equal to
an angle of the other, and the Jides about the equal angles reci-
procally proportional, are equal.
Let the triangles be {o placed that the equal angles
^^ and ^A may be vertically oppolite, that is to lay,
lb that ^mmmmi^m and — -^— may be in the lame
ftraight line. Whence alfo i and -aiM^MMM mull
be in the fame ftraight line. (B. i. pr. 14.)
Draw ■— — — , then
>
(B. 6. pr. I.)
(B. 5. pr. 7.)
(B. 6. pr. I.)
BOOK VI. PROP. XV. THEOR.
235
>
(B. 5. pr. II.)
II.
Let the fame conftruction remain, and
(B. 6. pr. I.)
and
A
(B. 6. pr. I.)
But — — . : ^— :: ; . ■ , (hyp.)
(B.5 pr. 11);
(B. 5. pr. 9.)
• • •
> -^
Q.E. D.
236
BOOK VI. PROP. XVI. THEOR.
PART I.
Y four Jh'ciight lines be proportional
the reSlangle ( ■
: ■■).
!■>..._.. ) contaified
by the extremes, is equal to the rectangle
X .........) contained by the means.
PART II.
And if the reSt-
angle contained by
the extremes be equal
to the reBangle con-
tained by the means,
the four Jlraight lines
are proportional.
PART I.
From the extremities of •— i» and "
^M^BHB and ————— _L to them and ^
draw
and ——•.-—- refpedlively : complete the parallelograms
^^^^H and
I
And fince,
• BB««»»«B»
(hyp.)
(conft.)
•
H (B. 6. pr. 14),
BOOK VI. PROP. XVI. THEOR. 237
that is, the redtangle contained by the extremes, equal to
the redangle contained by the means.
PART II.
Let the fame conftrudlion remain ; becaufe
• •
('
and 11 ^. -....-..■. ,
(B. 6. pr. .4).
But = ,
and — — i^ ^ — — — . ^conft.)
(B. 5. pr. 7).
Q. E. D.
238
BOOK VI. PROP. XVII. THEOR.
fince
then
PART I
F three Jlraight lines be pro-
portional (—1 : ^^^mmt
:: — — : ) the
reSlangle under the extremes
is equal to the fquare of the mean.
PART II.
And if the reSlangle under the ex-
tremes be equal to the fquare of the mean,
the three fir aight lines are proportional.
PART I.
Aflume
X
and
X
(B. 6. pr. 16).
or
But
X
■"9
- X
_.«i» ^ ; therefore, if the three ftraight hnes are
proportional, the redlangle contained by the extremes is
equal to the fquare of the mean.
Aflume •
PART II.
« , then
X
• «
• •
and
m» 9
^^ •
(B.
6. pr. 16),
• •
wmm
Q. E. D.
BOOK VI. PROP. XVIIL THEOR. 239
N a given Jlraight line ( ) ^,.
to conftruSi a reBilinear figure
fimilar to a given one (
and /imiiarly placed.
^
Relblve the given figure into triangles by
drawing the lines -«—--- and ••..••*•.
At the extremities of — — — ^ make
^ = Jb^^ and % = \J^ :
again at the extremities of ■
and ^^ = ^^\ : in like manner make
? = ^ ^"^ V = V •
make -^^ =:
Then
-v
is fimilar to
It is evident from the conftrudlion and (B. 1. pr. 32) tliat
the figures are equiangular ; and fince the triangles
W ^" w
are equiangular; then by (B. 6.pr.4),
and
240 BOOK VI. PROP. XVIII. THEOR.
Again, becaule ^^^ and ^^B are equiangular.
mm ** tt»afBffia«ai •
/. ex asquali.
(B. 6. pr. 22.)
In like manner it may be fhown that the remaining fides
of the two figures are proportional.
.-. by (B. 6. def. i .)
is fimilar to
and fimilarly fituated ; and on the given line
Q^E. D.
BOOK VI. PROP. XIX. THEOR. 241
I M I L A R trian-
gles (
A
and ^^^^^k ) are to one
another in the duplicate ratio
of their homologous Jides.
Let
^^ and A
be equal angles, and
and
homologous fides of the fimilar triangles
i^HHft and MKKKL
and ^^^^^^ ^nd on -.-.-.—— the greater
of thefe lines take --■— ■ a third proportional, fo that
* «■■■■■■■■■ \
draw
(B. 6. pr. 4) ;
but
(B. 5. pr. 16, alt.),
■■■>■■■
MiSa
(conll:.),
— confe-
1 1
242 BOOK VI. PROP. XIX. THEOR.
A\
quently ^^^ rz ^^ for they have the fides about
the equal angles ^^ and ^Ik reciprocally proportional
(B. 6. pr. 15);
Aa-A\
(B. 5 pr. 7);
^^^L : ^^ :: ....
(B. 6. pr. I),
that is to fay, the triangles are to one anotlier in tlie dupli-
cate ratio of their homologous fides
— i— and i^— -i (B. 5. def. 11).
Q^ E. D.
BOOK FI. PROP. XX. THEOR.
243
IMILAR poly-
gons may be di-
vided into the
fame number of
fimilar triangles, eachfimilar
pair of ivhic/i are propor-
tional to the polygons ; and
the polygons are to each other
in the duplicate ratio of their
homologous fides .
Draw
and
and
and " 5 refolving
the polygons into triangles.
Then becaufe the polygons
are limilar,
and —
■■■•■■««««■
and
♦=♦
are fimilar, and ^^ ^ ^J
(B. 6. pr.6);
but ^F^ = w becaufe they are angles of fimilar poly
gons ; therefore the remainders ^^ and ^k
hence nmmmmmmmmm * ■■>..■«•• \\ -_..._-__ *
on account of the fimilar triangles,
are equal ;
* ?
244 BOOK VI. PROP. XX. THEOR.
and --. : :: I
on account of the fimilar polygons,
■ ■■■■*■■■■» • _^.^_^_— — ___ •• ■■■■■■■HMM* • _^
ex asquali (B. 5. pr. 22), and as thefe proportional fides
contain equal angles, the triangles
s ^^^ and ^^^
are fimilar (B. 6. pr. 6).
In like manner it may be fhown that the
triangles ^^F and ^^K are fimilar.
^^ and ^^m
But -^^» is to ^^^m in the duplicate ratio of
..-■■..... to .—.—>— (B. 6. pr. 19), and
^^^ is to ^^
in like manner, in the duplicate
ratio of -.■•■•■■-.• to —-——.;
>>
(B. 5-P'-. II);
Again ^^^^ is to ^^^ in the duplicate ratio of
^^^ to ^^
to — ^— — , and ^^^F is to ^^r in
T
BOOK VL PROP. XX. THEOR. 245
the duplicate ratio of ^i^— to .
and as one of the antecedents is to one of the confequents,
fo is the fum of all the antecedents to the fum of all the
confequents ; that is to fay, the fimilar triangles have to one
another the fame ratio as the polygons (B. 5. pr. 12).
But ^^M is to ^^^F in the duplicate ratio of
to
Q ED
246
BOOK VL PROP. XXI. THEOR.
ECTILINEAR Jigures
(
<?«</
which are fimi/ar to the fame Jigure (
are fimilar alfo to each other.
Since HHiBll^ and are fimi-
lar, they are equiangular, and have the
fides about the equal angles proportional
(B. 6. def. i); and fince the figures
and '^%. are alfo fimilar, they
are equiangular, and have the fides about the equal angles
proportional ; therefore IHIBl^ and l^Hhk. are alfo
equiangular, and have the fides about the equal angles pro-
portional (B. 5. pr. 1 1), and are therefore fimilar.
Q,E. D.
BOOK VI. PROP. XXII. THEOR. 247
PART I.
Y four Jlraight lines be pro-
portional (^^^ I ^^—
:: — : ), the
Jiinilar reSiilinear figures
fimilarly described on them are aljo pro-
portional.
PART II.
And if four fimilar reSlilinear
figures, fimilarly defcribed on four
jlraight lines, be proportional, the
firaight lines are alfo proportional.
Take
and —
to
fince
PART I.
a third proportional to
, and —■••••■•• a third proportional
— > and — — — > (B.6.pr. ii);
:: ; (hyp.),
■ — ■ :: -— — : -■••••••••• (conft.)
.*. ex asquali.
but
and
(B. 6. pr. 20),
248 BOOK VI. PROP. XXII. THEOR.
and ,*.
(B. 5. pr. 11).
PART II.
Let the fame conftrudlion remain
(B. 5. pr. II).
(hyp-).
(conft.)
(^E. D.
BOOK VI. PROP. XXIII. THEOR. 249
QUIANGULAR parallel-
ograms ( and
^m^ ) are to one another
in a ratio compounded of the ratios of
their fdes.
Let two of the fides
and
-«... . about the equal angles be placed
fo that they may form one ftraight
line.
Since ▼ 4. M -- f\\ ,
and 1^^ = ^W (hyp.).
and .*.
+
and
form one flraight line
(B. I. pr. 14) ;
complete ^ ,
Since
#
• ■■«■•
and
#
(B. 6. pr. i),
(B.6. pr. i).
has to
- to .„.
a ratio compounded of the ratios of
, and of ^^— — to — n^— »» .
K K
Q^E. D.
250 BOOK FT. PROP. XXIV. THEOR.
-B
N any parallelogram (^7 /)
the parallelograms ( r^i
and ^ I ) 'which are about
the diagonal are Jimilar to the whole, and
to each other.
B-J ^^
As ^ I and ^ I have a
common angle they are equiangular ;
but becaufe
and
are fimilar (B. 6. pr. 4),
and the remaining oppofite fides are equal to thofe,
, It 1 and fn I have the fides about the equal
angles proportional, and are therefore fimilar.
In the fame manner it can be demonftrated that the
parallelograms ^7 / and ^ / are fimilar.
Since, therefore, each of the parallelograms
B ..^E
^.
is fimilar to ^1 I , they are fimilar
to each other.
Q. E. D.
BOOK VI. PROP. XXF. PROB.
251
O defcribe a reSlilinear Jigure,
ivhic/i /Jiall be Jimilar to a given
reBilinear Jigure (
equal to another (^^ ).
), and
Upon defcribe
i_ defcribe | | = ^^,
and upon «_
and having ^M ^
(B. I. pr. 45), and then
smm
Between
and
■H»» will lie in the fame flraight line
(B. I. prs. 29, 14),
and nu»»H.. find a mean proportional
(B. 6. pr. 13), and upon _^««_«i
defcribe Jtt^ 9 iimilar to
and fimilarly fituated
Then
For fince
and
are fimilar, and
(confl.),
■ ■■■■■■■■■
(B. 6. pr. 20) ;
252 BOOK FI. PROP. XXV. PROP.
but 1
• •
but .^d^k = ■
(B.6.piM);
(B.5.pr.ii);
and .♦.
(conft.),
(B. 5. pr. 14);
I
and
(conft.) ; confequently.
which is limilar to
is alfo =
Q. E. D.
BOOK VI. PROP. XXVI. THEOR.
253
F fitnilai' and Jimilarly
pojited parallelograms
have a common angle, they are about
the fame diagonal.
For, if poffible, let
be the diagonal of
draw ■
(B. I. pr. 31).
Since
P.. ^
are about the fame
and have
diagonal ^^^^^^^^ , ana nave jmm common,
they are fimilar (B. 6. pr. 24) ;
but -
(hyp.).
and .*.
(B. 5. pr. 9.),
which is abfurd.
3
is not the diagonal of
in the fame manner it can be demonftrated that no other
line is except : .
Q. E. D.
254
BOOK VI. PROP. XXVII. THEOR.
F al/ the reBangles
contained by the
fegments of a given
Jlraight line, the
greateji is the fquare which is
defer ibed on ha f the line.
be the
unequal fegments,
equal fegments ;
For it has been demonftrated already (B. 2. pr. 5), that
the fquare of half the line is equal to the redlangle con-
tained by any unequal fegments together with the fquare
of the part intermediate between the middle point and the
point of unequal fection. The fquare defcribed on half the
line exceeds therefore the redtangle contained by any un-
equal fegments of the line.
Q.E. D.
BOOK VI. PROP. XXFIII. PROP.
^SS
O divide a given
Jlraight line
fo that the rec-
tcuigle cojitained by its segments
may be equal to a given area,
not exceeding the fquare of
half the line.
Let the given area be :=
Bifedl —
or
make
and if
But if
muft
■ ■■■
» **** ymmmm***
■■■1
•;
. 2
the
problem
is folved.
9
■II ^
-4;- ••■»■■■■••■
9
then
■■■
(hyp.).
Draw
make -
with ^-i
■ ■■■■IMHM I
or
as radius defcribe a circle cutting the
given line ; draw
Then •••— ^ wiMMaBB.-a- ^
(B. 2. pr. 5.) =
.2
But
+
(B. I. pr. 47);
256 BOOK VI. PROP. XXVIII. PROB.
.\ X — ■ + ^—
= ' + \
from both, take — i"— ^— ^^
and ""■■ X ■— — ^^—•■••« S3 «MMB
But "■■ ' ■ ' =: — — — •• (conft.),
and /. ■-■"" is fo divided
that •"•"• X ——————— ^: —•••.-2^
Q^E. D.
BOOK VI. PROP. XXIX. PROB.
^S7
O produce agivenjlraight
line ( ), fo
that the reBangle con-
tained by the fegments
between the extremities of the given
line and the point to which it is pro-
duced, may be equal to a given area,
i. e. equal to the fquare on
Make
draw -—"--■
draw
with the radius
meeting
Then —■■-—'
•-, and
But
and
5 and
', defcribe a circle
■ produced.
' (B. 2. pr. 6.) = —
-^ +
(B.i.pr.47.)
:= the
given area.
Q^E. D.
L L
258
BOOK VI. PROP. XXX. PROB.
1
O cut a given finite Jlraight line ( ■ — ■• )
in extreme and mean ratio.
On
defcribe the fquare
I
(B. I. pr. 46) ; and produce
X "
, fo that
^ s
(B. 6. pr. 29);
take
and draw ^
meeting ^
Then
U
■•aatB •■
X-
;B. I. pr. 31;
■ ■■■taaBB
and is .*. ^
n
; and if from both thefe equals
be taken the common part
\ I , which is the fquare of ■
will be = ■ , which is = '■■- X
■ ■««■■«■*« *
that is
and
.■ is divided in extreme and mean ratio.
(B. 6. def. 3).
CLE. D.
BOOK FI. PROP. XXXI. THEOR. 2
59
F any fimilar reSlilinear
figures be fimilar ly defer ibed
on the fides of a right an-
gled triangle ( ^''•»^ ), the figure
defer ibed on the fide (■....i ) fuh-
tending the right angle is equal to the
futn of the figures on the other fides.
From the right angle draw
to •> m.
then ■■■■■•■■■MHB : ^_— i—
perpendicular
but
(B. 6. pr. 8).
(B. 6. pr, 20).
(B. 6. pr. 20).
• •ammmmmm%mmm
Hence
but
+
+
and /.
Q. E. D.
26o BOOK VI. PROP. XXXII. THEOR
F two triangles ( ^ ^ ^«^
/^\ ), have two fides pro-
portional ( ..1.^.^^ : I I
\\ .. .......... I •.•••..•••.), and be fo placed
\ i7^ an angle that the homologous Jides are pa-
rallel, the remaining Jides (
one right line.
and
) form
Since
= (B. I. pr. 29);
and alfo fince -^— ^ || ••■••••■>■
= ^^ (B. I- pr. 29);
= ^^^ ; and fince
■ ■■■«■■■««■ •
— (hyp.).
the triangles are equiangular (B. 6. pr. 6) ;
M = /S
but
A+ +A =
+
A
+ JI =
■*■«■»■»«
I I 1 (B. I. pr. 32), and /. -*••—■» and
lie in the fame ftraight line (B. i. pr. 14).
Q,E. D
BOOKVL PROP. XXXIII. THEOR. 261
N egua/ circles (
O-O
), angles.
whether at the centre or circumference, are
in the fame ratio to one another as the arcs
on which they Jland (
fo alfo are fedlors.
i-J::-
o
)■>
Take in the circumference off 1 any number
of arcs "■— ■ , ■— , &c. each ^ ^m» ^ and alfo in
the circumference of f j take any number of
arcs • , , Sec. each ^ •***«*•, draw the
radii to the extremities of the equal arcs.
Then fince the arcs — , —. , i..., &c. are all equal,
the angles # , # , ^, &c. are alfoequal (B. 3. pr.27);
.*. ^V is the fame multiple of 0 which the arc
is of ^1^ • and in the fame nianner ^Bi^
is the fame multiple o
is of the arc
which the arc
.... •>••
V*
262 BOOK VI. PROP. XXXIII. THEOR.
Then it is evident (B. 3. pr. 27),
if" ^11^ (or if m times w ) C> => ^ Mfg^
I
(or n times ^ )
then ^^fc_i,i«^^ (or »; times '••••^) C!> ■
.....^••* (or n times ) ;
.... , (B. 5. def. 5), or the
angles at the centre are as the arcs on which they fland ;
but the angles at the circumference being halves of the
angles at the centre (B. 3. pr. 20) are in the fame ratio
(B. 5. pr. 15), and therefore are as the arcs on which they
ftand.
It is evident, that fedlors in equal circles, and on equal
arcs are equal (B. i. pr. 4; B. 3. prs. 24, 27, and def. 9).
Hence, if the fedors be fubftituted for the angles in the
above demonftration, the fecond part of the propofition will
be eftablifhed, that is, in equal circles the fedlors have the
fame ratio to one another as the arcs on which they ftand.
Q^E. D.
BOOK VI. PROP. A. THEOR.
Y the right line {'mmm^um,),
bifeSling an external
angle ^H of the tri-
yf
263
angle
z.
meet the oppojite ^
Jide (-^^— •) produced, that whole produced fide ( "■■•),
and its external fegment (——--—) will be proportional to the
fides (-^— — ■..— and ), which contain the angle
adjacent to the external bifeSled angle.
For if I be drawn || -.---»•»■• ^
then ^^ = \ / , (B. i. pr. 29) ;
and
= ^,(hyp-).
= ^P, (B. I. pr. 29);
r........ zz. ■ I III. , (B. I. pr. 6),
and
(B. 5. pr. 7) ;
But alfo.
■ •■■■■■■■■■■ ,
(B. 6. pr. 2);
and therefore
(B. 5. pr. I,).
Q. E. D.
264
BOOK VI. PROP. B. THEOR.
X
F an angle of a triangle be bi-
Je5ied by a Jlraight line, which
likewife cuts the bafe ; the rec-
tangle contained by the Jides of
the triangle is equal to the rectangle con-
tained by the Jegments of the bafe, together
with the fquare of the Jlraight line which
bifedls the angle.
Let
be drawn, making
^ = ^; then fhall
X +
^r \ defcribe I
About y \ defcribe J (B. 4. pr. 5),
produce 1 ■■ to meet the circle, and draw ■■■»>—
Since ^ = ^^ (hyp-)'
and ^^ = ^ (B. 3. pr. 21),
• * •C*i»l»l
ind
\
are equiangular (B. i. pr. 32) ;
(B. 6. pr. 4)
ROOK FI. PROP. B. THEOR. 265
(B. 6. pr. 16.)
.- X + '
(B. 2. pr. 3);
■*«■«■
but X — — = X
(B- 3- pr. 35)'
X = X
Q.E. D.
MM
266
BOOK VI. PROP. C. THEOR.
fhall
Y from any angle of a triangle a
Jlraight line be drawn perpendi-
cular to the bafe ; the rectangle
contained by the fdes of the tri-
angle is equal to the reSlangle contained by
the perpendicular and the diameter of the
circle defcribed about the triangle.
From
draw ■«>ii»«afa««
.. X
of ^y
..-•
— ; then
Xthe
diameter of the defcribed circle.
Defcribe
O
(B. 4. pr. 5), draw its diameter
and
. and draw ^-im— • then becaufe
^ ■ >• (conft. and B. 3. pr. 31) ;
,Xl = /> (B. 3. pr. 21);
.%*<
is equiangular to / ^
• MHHaHMMHM
l/^
and ,*. .-— .—
(B. 6. pr. 16).
(B. 6. pr. 4);
X
Q^E. D.
BOOK VI. PROP. D. THEOR.
267
IHE reStangle contained by the
' diagonals of a quadrilateral figure
I infcribed in a circle, is equal to
\ both the reBangles contained by
its oppoftte Jides.
/ /be any quadrilateral
/
o
fieure infcribed in
and draw
and
then
X
X
Make
^k = W (B.i.pr. 23),
^ = ^ ; and
(B. 3. pr. 21);
= 0
«■■■■■■!■■«
(B. 6. pr. 4);
and ,*.
X
X
(B. 6. pr. 16) ; again,
becaufe ^^ ^ ^F (conft.),
X-
■■■■■•■••
268 BOOK FL PROP. D. THEOR.
and\/ = \^ (B. 3. pr. 21);
•■«■■■«•■■• * ■■■■■*>■■■
THE END.
(B. 6. pr. 4);
and ,'. '•"••■-••• ^ .^__^ ^ .•■■•■•■■■■• ^ a^^^MB
(B. 6. pr. 16) ;
but, from above,
X = X ;
— = X + X
(B. 2. pr. I .
Q^E. D.
cHiswirK: PRiNirn by c. " iirxTiNoinM.
fuciJ
■O.,'
^ \'yj
Live Music
Archive
Librivox
Free Audio
Metropolitan Museum
Cleveland
Museum of Art
Internet
Arcade
Console Living Room
Open Library
American
Libraries
TV News
Understanding
9/11